The essence of a D&D game is its adventures, and the core of an adventure is its encounters. Encounters are what stand directly in the way of the PC accomplishing whatever it is they want to do. It's possible to have obstacles that aren't encounters (like the core revelation of a mystery story), but traditionally, encounters form the bulk of the obstacles in a D&D game. That doesn't need to change for DM+1 play, but the way that you think about how to set up encounters might.
On the plus side, remember how I'd said before that DM+1 play has the advantage of being more in line with the single protagonists of most inspirational media? This means that it's usually easier to take those cool moments and inject them into a DM+1 game than doing so with a larger group.
But I'm getting ahead of myself. I think most encounters belong in (at least) one of four categories: monsters, traps, tricks, and puzzles. I'll go through each in turn to define what I actually mean by it and how I'd approach it (in general, and where there's more to say, for a DM+1 game), including an example, with section headers since this turned into a lengthy post.
(Note: as before, I'm using the term "monster" to refer to any NPCs)
Monster encounters are the simplest to define, since it's any time the PC is obstructed by an NPC. Note that I'm not mentioning anything about combat, subterfuge, social interaction, or any other form of resolving the obstruction. All the GM should do is bring the encounter into the game; how to handle it is up to the player to decide. This doesn't mean the NPC can't initiate action (in my last DM+1 session, the PC had separate encounters with an animate statue guardian and an ooze creature, both of which attacked her upon detection because it made sense for them to do so), just that the GM shouldn't impose arbitrary restrictions on an encounter unless they're sensible within the game's internal logic.
Putting together monster encounters is simple enough on the surface (just chuck a few NPCs in there and play out what happens next), but I've got a few further guidelines to help make them fit better with DM+1 play.
When it comes to selecting monsters, they tend to fall into three broad categories: intelligent (on at least a near-human level, with capacity for complex emotions, motives, desires, fears, and communication), unintelligent (animals and the like, capable of instinctive thought but limited complexity), or mindless (beings with little to no self-awareness and individual capacity, such as stereotypical zombies or golems). Of course, combinations are also possible, such as bandits (intelligent) who make use of hunting dogs (unintelligent) or a necromancer (intelligent) with a personal guard of skeletal warriors (mindless). Generally speaking, intelligent monsters tend to be preferable since they present more avenues for interaction and play, but there's nothing wrong with using the others as a change of pace. I'd suggest at least a 2:1 ratio of encounters with intelligent monsters to those without, but that's a broad guideline that I wouldn't hesitate to violate in a case where having so many intelligent monsters wouldn't make sense (e.g. a haunted valley where the dead reanimate each night to play out an ancient battle).
On a related note, I strongly recommend use reaction rolls (I tend to stick with the B/X 2d6 standard since I know that table by rote, but this 2d10 alternative from The Dragon's Flagon is fine if you worry about Charisma having an excessive effect with universal stat modifiers) unless there's a reason for the monsters to have a certain initial attitude. Not only can it lead to interesting situations, but it helps avoid screwjobs (real or perceived) because the player's choices will tend to be the leading cause of violence.
I've seen suggestions for adjusting monster stats for smaller numbers of PCs (such as this one by Justin Alexander), but those tend to fiddle too much with math for my tastes. To handle it on the fly, I just treat the PC's damage as affecting Hit Dice instead of hit points, treating any extra static bonus as +1 Hit Dice (e.g. if an attack would normally deal 3 points of damage, it'll kill a monster with less than 3+1 Hit Dice or knock a standard OD&D troll [with 6+3 Hit Dice] down to just above half health). Any special defenses affecting damage received (e.g. a standard OD&D wraith taking half damage from silver-tipped arrows) work as normal. If it's supposed to be a notably tough monster, they can make a save vs. death to resist a fatal hit, modified by any overflow damage (e.g. if the previous example's troll was then hit for another 5 damage, it could try to make a save with a -1 modifier to stay at 1 Hit Dice). This might sound like it would lead to more reckless play, but I haven't run into that issue in practice; the PC is always vulnerable enough that players tend to opt for non-violent approaches (or at least approaches that keep them personally insulated from direct violence, like using charmed minions) as much as possible.
Lastly, take note of what monsters are in the area, whether keyed or as random encounters, and seed hints about their presence where reasonable. My practice of late has been to do this as part of random encounter checks (e.g. for a 1-in-6 check, a d6 roll of 2 results in finding traces of a monster and a 3 results in hearing noise/noticing smells/etc. from a nearby monster), but regardless of whether you prefer to do that or include such hints in keyed descriptions, adding touches of foreshadowing like that can help the player make more informed choices about their actions along with feeling satisfying on a meta-level when the monster shows up later.
As a worked example, consider the giant spider in chapter two of the Conan story The Tower of the Elephant. For simplicity, I'll use the stats of a standard B/X giant black widow spider. This beast has presumably been compelled by magic to act as a guardian for the tower's top (so it wouldn't need to do any reaction roll), and there's no need to roll for the number appearing since the story is explicit that there's only one. It takes two blows from Conan in the story; a sword strike to its legs as it runs by early on, and then a hit from the thrown chest; and 3 Hit Dice is on the low side, so I'd allow it to make a save to resist a fatal hit. That's it! Everything else about it can be run as if it was any other encounter in a D&D game.
Traps are situations with predetermined triggers (generally autonomous or simply manipulated) and effects (always meant to be negative for whoever triggers them). They can be mechanical (like the pressure plates before the golden idol in Raiders of the Lost Ark) or magical (like the Midas statue's hand in Tomb Raider), with effects ranging from simple damage and save-or-die to blowing out light sources, causing monster encounters, sealing exits, altering passages, transportation/teleportation to a worse place, or anything else that can be imagined.
I'm an unabashed fan of traps. When used well, not only are they a fun intellectual challenge for the player, but I think it's a mark of a great player when they can figure out how a trap works and turn it around on enemies. That said, there are lot of ways that traps can be used poorly, too.
The first key is for the GM to have a clear idea of the trap's trigger(s) and effect(s) (for simplicity, I'll stop doing that going forward). Courtney Campbell's Tricks, Empty Rooms, & Basic Trap Design PDF is a great resource for this, as is the trap/trick index on his blog. How one actually relates to the other isn't as important, generally speaking. Think of it if you can, by all means, but it's very rarely worth stressing out over understanding exactly how a pressure plate sets off a dart launcher, for instance. Even in a case where the mechanism can be exposed and compromised/disabled, just saying that is often good enough, rather than delving into the details of it.
Once those are set, use them to figure out how to hint at the trap's presence. This can be obfuscated (by mentioning it alongside other details in the room's description or requiring some initial investigation to discover, for instance), but it should never be skipped. If a trap can't be detected in some reasonable way before setting off its trigger, it's a bad trap (how subtle a GM can be while remaining reasonable depends on the metagame context, but a rough guideline is that it should seem a bit obvious to you, because such clues are always harder to recognize from the player's side). For instance, with the gas traps in the chapel of evil in S1: Tomb of Horrors, the PC should be able to notice some seal on the seams of the trapped covers (if the trigger is simply opening them), to find something like a tripwire (if opening them ruptures a compressed gas container), or to see some hint of arcane scrawling on the underside of the cover where it protrudes from the base (if opening them conjures the gas). If a trap is effectively undetectable, it comes an arbitrary tax on the player, which is just poor design.
This can be easy to overlook, but make sure to locate the trap in a sensible place (assuming it wasn't set up by a crackhead wizard). Traps with mechanical triggers in areas with monster traffic are usually not sensible unless the monsters are entirely incapable of triggering them (e.g. a neck-height tripwire in a den of short monsters or pressure plates on the ground of a beholder's lair). Those with effects that capture instead of harming might have a little more latitude for this, but give some thought to how annoying it'd be to keep having to free allies (from the monsters' perspectives) from it. Traps with magical triggers tend to have more leeway for selective targeting, but make sure to think of the details in order to be fair in how it's adjudicated during play. In general, traps are best placed in locations where the inhabitants can't set them off by accident, but exceptions can be made depending on how cheaply they view life.
In fact, that can be extended to a general guideline of make sure the trap makes sense within the game (this idea also applies to any other encounters).
Decide if the trap is a one-off occurrence or capable of repeating, with an idea for how it gets reset in the latter case (e.g. is it just a matter of time, will a random encounter result correspond to a monster resetting it, etc.). This tends to be straightforward, once the trigger and effect are set, together with the overall context and intent of the trap.
For thoughts on effects from traps other than straight damage, see what I said about special abilities for monsters in DM+1 games in my previous post.
There's more to traps that just those pointers, but those should serve well to get started out. Most of the rest of what I can say about traps would be situational details and nuance, which is better learned through practical experience.
As a worked example, I'll revisit what I'd mentioned in my previous post and recreate one of the tentacle drag hallways from Dead Space. The set-up is to have a lengthy hallway with an aperture, beyond which dwells some tentacled beast (I'm imagining it reaching with something that looks like an elongated lobster tail on the end of a thick crab leg). The trigger is the beast detecting prey within reach of its tentacle, but that begs figuring out what would it consider prey. For our purposes, I'll say it considers anything from about the size of a medium dog (like a Labrador) up to the size of a large human (like Shaq) as prey and that it's limited to sensing sounds and ground vibrations (to allow for a non-straight hallway). The effect is getting snatched by the tentacle and dragged to its mouth, where its beaked maw bites for high damage each round. If grabbed, the PC can break free by causing some amount of damage to the tentacle (among other means).
Before going further, I want to define the mechanics more clearly, for the selfish reason of this being something I'd like to use. Since D&D isn't limited by 3D model animations, the tentacle should be able to grab multiple locations, which can be simulated by a 1d10 roll:
1-2 = Left leg
3-4 = Right leg
5-7 = Torso
8 = Left arm
9 = Right arm
10 = Head
I want the drag to take long enough that the player has time to make one or two mistakes, and I also want them to need to put in more than two rounds of effective actions to break free, so taking 1d4+3 rounds sounds reasonable, off the top of my head. In fact, I can also throw in a saving throw to increase the time by avoiding a solid initial grab, so I'd say a save vs. breath can increase the drag time to 1d4+5 rounds. The actual dragging speed would matter if I was putting this into a mapped hallway, since that could interact with other features in the hall, but I can skip that to keep it simple for now. Since I tend to play at levels 1-5, I'll say the beast's bite deals 6d6 damage; it's probably likely to be lethal (depending on the PC's hit points) since there's about an 80% chance of 16-26 damage per bite, but there's a chance of surviving one bite for a final desperation action. For the damage needed to break free, I'll say it has 5 Hit Dice, with unarmored Armor Class for its weak spot (perhaps a cartilage joint at the change of its carapace, requiring missile attacks to reach) or chain mail Armor Class for its general mass. I'd consider doing some statistical analysis to see how deadly this is, but those numbers are good enough for a first blind stab.
Getting back on track, I need to come up with some clues about the beast. Intelligent inhabitants could've put up some warning signs or installed a simple barricade beyond the tentacle's reach. There could be a conspicuous absence of vermin or minor wildlife nearby. There could be drag marks on the floor, blunt impact damage on the floor and walls, or crushed paths in foliage, depending on the setting. There could be pieces of the tentacle carapace that have been chipped off previously. There could be a rhythmic clicking or an alluring scent that the beast uses to lure in prey. There could be a pile of remains near the aperture, giving the hallway a stench of decay.
Making this make sense within the setting seems pretty straightforward to me, especially if there are alternative paths between the areas at either end of the hallway that would be used by intelligent inhabitants. Aside from noting that the aperture should probably connect to other possible feeding grounds, I can leave this to figure out when I've got a dungeon to use this in.
Escaping the tentacle by damaging it would yield a lengthy reset time (more if the damage was dealt to its weak spot). Escaping by other means would probably just give a brief reprieve until it's ready to strike again, depending on the exact method.
Tricks are pretty similar to traps, except that the effects are designed to mislead, misdirect, or fool the player, and they can be persistent environmental effects rather than needing to be triggered. Think of something like the arthropod hall from Indiana Jones and the Temple of Doom; the vermin are disgusting and off-putting, and they hide a clear view of the release mechanism, but they're ultimately harmless (unless perhaps the PC has a terminal zoophobia). In fact, I'd say that one doesn't really need any specific mechanics to recreate it in a D&D game; the GM just has to play up how creepy, gross, and/or unnerving the environment is.
As with traps, think of what's there, why it's there, and what the consequences of its presence would be. There really isn't anything else to be said about them without resorting to talking about specific tricks.
On which note, for a worked example, consider Talia's death scene from the 2016 Blair Witch movie, from the perspective of either Sara or James as the PC (there's additional context in the movie, but I can work around it for this example). The PC is trying to find an Eldritch Location within a forest haunted by a supernatural being they want to avoid, they've got an NPC friend who's coming along as an ally, and they've hired an eccentric local guide. During the journey, the guide acts strangely, fomenting some strife with the ally if not also with the PC. The following morning, there are a handful of twig effigies hanging above the campsite that nobody witnessed being put up, but some investigation shows that they were bound with fabricated twine of the same sort that the guide has. Tensions mount, but the PC needs the guide's help to reach their destination, so they find a way to reach a truce and press on. The next morning, there are strangely regular piles of stones set up, and dozens if not hundreds of effigies hang around the campsite, ranging in size from small twigs to huge tree trunks. The guide begins to have a breakdown upon noticing one of the effigies has a tuft of their hair on it, and their mewling begins to visibly upset the ally.
If the player asks or wonders aloud about it, the GM can agree that it would've been almost impossible for either NPC to have staged the event. The new effigies could be bound by natural fibers like cured vines, sinew, or gut, rather than fabricated twine. The guide's hair might not have any detectable place where the tuft was cut. There could have been rumors in the town about signs of the being's attention including the effigies and stone piles.
Play up the ally's increasing anger, and if the PC doesn't take action to intervene and diffuse the situation, the ally snaps the tufted effigy, killing the guide inadvertently. Now the PC and the ally will have a far harder time getting anywhere, on top of potentially roleplaying through the ally's guilt and trauma.
Puzzles are essentially some mixture of tricks and traps, with more possibility for an element of being spread out over time and a focus on being intellectual challenges. There are two broad categories of puzzles: those take make sense within the game's world and those that are meant to be solved at the metagame level. Personally, I have low tolerance for the latter, so I'm not going to talk about them directly, though I suppose much of the following could still be applicable to them.
Before I go any further, let me say that a puzzle isn't worth putting into a D&D game if it can be "solved" with a simple dice roll. Doing that will be unsatisfying to a player who enjoys solving puzzles, and just removing the puzzle is a better solution for a player who doesn't enjoy solving puzzles than inserting a crutch mechanic that relies on random chance.
Puzzles can come in a variety of forms, from a single-room challenge like the "name of God" scene from Indiana Jones and the Last Crusade to adventure-defining mysteries like figuring out where to find a galaxy on Orion's belt in Men in Black. The following is aimed at smaller-scale puzzles, but I'll note where any particular issues or considerations are needed to apply them on a larger scale.
The most important thing is for the puzzle to make sense as an alternative to whatever other security measures could be taken if it was intentional (excepting cases like being set up as a challenge by a crackhead wizard) or to have a sensible reason for why it hasn't been resolved if it was coincidental. Personally, I tend to lean in favor of problems with no immediately obvious solution (to the point where they can essentially turn into complex tricks/traps) rather than the sorts of things that come up in brainteaser collections. A magical Sudoku puzzle sealing a vault door tends to break my suspension of disbelief and jar me out of the game. A moat full of piranhas surrounding an oversized treasure claimed by a hostile creature that can fly, on the other hand, is far more likely to get me interested and engaged. This concept scales up pretty easily, eventually morphing into checking for plot holes on a mystery when the puzzle is the entire adventure.
If there's any critical contextual information that's needed to understand the parameters of the puzzle, plan for ways of sharing it (not exhaustively, but I'd say there should be at least two or three ways of getting to any critical information). The "name of God" scene would fall flat without having books or scholars where it could be found (scholars being more common but relying on either oral descriptions or modern translations keeps that further element of puzzle). A puzzle involving shining red light on a green portion of a stained glass barrier to disable it wouldn't make sense without some source of color theory (like a painter or a mythological muse). Wordplay is generally acceptable without breaking suspension of disbelief if it's in your main language at the metagame level (or if it's in a real world language that's in the game world explicitly), but I'd suggest core elements rely more on either scientific (like the true spelling being "Iehova") or poetic logic (like Orion the cat having a collar as a "belt") where possible.
Don't try to force a single solution on the puzzle (or, on a larger scale, don't try to force the player's progression to fit a preconceived plot). It's probably worth thinking of two or three approaches that might resolve the puzzle to make sure that the player will have options, but as I've said before, be open to working with what the player comes up with. This is especially important if the player is using up some limited resource, because it can be very frustrating to use up an item or a spell only to be told it does nothing. This is what tends to require the greatest break from inspirational material, since creators of linear fiction have the benefit of being able to contrive exactly the solution that they want.
Be very careful about using a puzzle that will halt progress until it's solved. I'm not saying this should never be done, but it should be a conscious and intentional choice when done, because it becomes the puzzle-equivalent of a final boss monster. If you're not sure about it, think of at least one way to bypass the puzzle to make sure it isn't blocking progress along the adventure's critical path. This is what contends with the previous point for requiring the greatest break from inspirational material.
Lastly, this isn't necessary, but I find most puzzles benefit from having some element of time pressure, beyond which they become unsolvable or solving them becomes a different challenge. Removing a glowing crystal discharges the stasis field around some treasure, but now the PC has to fish it out from the bottom of a vat of acid before it corrodes away. The creature that nests across the piranha-filled moat will return in an hour and is too powerful to fight directly. The kidnappers must be caught before they collect enough sacrifices to summon a demon. The wrongly-accused spouse will be executed in a week if the PC can't clear their name.
For a worked example, I'm going to go beyond mere encounter design and consider taking inspiration from the movie Pitch Black. The PC is teleported (perhaps as part of a different trap/trick) to a seemingly abandoned and largely barren location. First, they're faced with the simple tasks of finding sustenance and shelter. Assuming they're able to do that, they discover some hostile novel creatures that stick to the darkness, records indicating untold numbers of them live in the sunless places nearby, and an automated astrolabe counting down the remaining time to and duration of the next eclipse. They need to find a way of escaping. Good luck.
Of course, this would require more to fill in information gaps and to come up with potential means of escape, but that's basically designing a whole adventure, if not an adventure path, which is beyond what I want to get into with this post.
Putting It All Together
It's probably obvious, but there's no need for encounters to stick to just one of these ideas. Using monsters with a trap/trick can increase their threat level dramatically and/or provide creative means of dispatching them. I mentioned already that the initiating event for that example puzzle could be the effect of a trap/trick. Monsters that can't be fought without special/mythical means are like mobile puzzles. So on and so forth.
In terms of how to balance them, I always suggest GMs who're new to DM+1 play start off favoring tricks and puzzles. Those require the least amount of change to adapt to having fewer players. As you get more comfortable with the format, introduce more monsters and traps. My ideal ratio in general play is usually about 4 monsters:1 trap:1 trick:1 puzzle, but I usually aim for something closer to 2:1:1:1 or 2:2:1:1 for DM+1 play for two reasons. First, attrition from combat is magnified when there's only one PC, at least at a tactical level if not also at a strategic level (and it probably should both in most cases). Second, DM+1 combat is vulnerable to falling into a "your turn"/"my turn" mentality, which feels boring. Even with monster encounters not leading to combat in the majority of cases, I like to play up the other ways of challenging the player so that it feels more special when a combat situation does come up.
That said, I've had success running a variety of published adventures in DM+1 play without adjusting the encounters significantly, so it's not a must to stick to any special distribution. Put good content in the game, (try to) find good people to play it with, and run it in a fun way. You can find your own style from there.