I happen to be running a singleplayer campaign for my wife, so I feel qualified to offer some input.
First of all, I don't think every class is equally equipped to be a fun singleplayer PC, or at best each class requires a different type of gameplay focus to be enjoyable. A Fighter would probably be the all-around best choice due to its durability, high damage output, and being short rest-focused rather than long rest-focused. A Rogue, like what my wife plays, would do very poorly in a conventional dungeon crawl game (except maybe a Swashbuckler Rogue, since they work well solo by design), but is great for a game focused on exploration, social interaction, and skill challenges.
As such, a lot of advice depends on which class your player is going to play. If he plays a class that is less independently durable in combat, you might have to put in time to alter encounters to have different methods of resolving them, which requires you to think of something that is as
interesting and fun as combat, without relying on combat.
As far as optional rules, here are the ones I use, and whether I recommend them:
- Inspiration can be used to reroll after seeing the result (see DMG pg. 240, under "Heroism"). This helps a lot with survivability. And give your player Inspiration for doing anything cool/clever or for playing to his character's traits, especially negative ones. The fewer players you have, the more you can focus on individual character development and that character's "arc" and inner conflicts.
- Average damage instead of rolling damage for monsters. This prevents unexpected damage spikes that can suddenly take out a PC, which is fine in a normal-sized group but puts the brakes on the fun in a solo party. If you don't mind more risk, you can still do critical hits by rolling the damage dice associated with a normal hit and then adding the average damage (per DMG p. 248, "Monsters and Critical Hits")
- Hero points (DMG p. 264). I actually don't use this one right now, but I might choose to later. Should help your single player be more durable in combat and solve more challenges on his own, since he won't have as much variety in skill proficiencies as a full party.
- Healing surges (DMG p. 266). This rule is designed for parties with little or no healing magic. The PC regains all Hit Dice after a long rest, so he won't be forced to take a day or more off of adventuring because of a battle turning against him and making him use all of his Hit Dice. You should probably go with the suggestion of letting the PC use this option as a bonus action rather than his full action (I let my player do it as a bonus action, and battle still gets very intense)
- Rest variants (DMG p. 267). I use "gritty realism" because it fits the style of campaign (interpersonal intrigue and gradual buildup of the PC's influence over time), but you might consider using "epic heroism" instead. If you use this variant and run a pre-written campaign as written, then the PC should be able to survive more encounters per day. If you are running your own campaign and using the default guideline for encounters per short and long rest, it doesn't matter which variant you use because the math will work out the same.
- Initiative variants (DMG p. 271). If you only have one player, then using a fiddly variant like "speed factor" won't slow things down too much and makes combat more tactical, which is helpful when you only have one PC and a small number of enemies. I don't use this, but I've thought about it.
- Cleaving through creatures (DMG p. 272). This lets you use more, weaker monsters without totally screwing over your player, since he can kill multiple enemies in one attack (helping with the action economy problems).
- Lingering injuries (DMG p. 272). Consider rolling on this table when your player gets to 0 HP. Instead of dying, he is knocked unconscious and taken prisoner or left for dead, but he wakes up 1d4 hours later with 1 HP and a lingering injury from the table.
- Massive damage (DMG p. 273). Very helpful for PCs that deal a lot of damage but are more fragile (such as "alpha strikers" like Rogues. I chose this one in particular because she plays an Assassin Rogue, but by default it's very hard for an Assassin to actually assassinate anyone with how HP increases with enemy CR). I apply it selectively, so I don't usually make the player make system shock rolls (not that it usually comes up, since enemies don't usually cause enough damage compared to a PC's max HP), nor do I have solo "boss" enemies roll for system shock.
- Morale (DMG p. 273). I haven't used this much, but it can be useful for letting you still have combat with multiple enemies but letting the player "win" by doing things other than killing every single enemy, such as targeting a group's leader or injuring the enemies enough to make them run away rather than fighting to the death.
I hope those suggestions help. I really don't think you need to make such drastic houserules as letting the PC get 4 turns per round, since there are already small tweaks you can make that make a single PC much more heroic and formidable in the face of multiple opponents.
As far as the suggestions to add NPC party members, I have had a lot of success with that in my campaign. You can treat the campaign like a story-focused RPG like Neverwinter Nights or Mass Effect, where the player has access to take one of several interesting pre-generated party members wherever he goes. Even adding a single extra PC of the same level allows the player to fight much more difficult encounters than he would otherwise. It helps to make the NPCs complementary to the PC: if he's a Rogue, give him some Fighter, Paladin, or Barbarian NPCs to choose from to help draw enemy attacks while he uses their presence to get Sneak Attack; if he's a Fighter, give him a Cleric or Druid to heal and buff him.
You should try to keep the NPCs simple so that the player can switch between them easily without having to thoroughly learn the ins and outs of each class: for a Fighter, make it a Champion or pick a Battlemaster with a few easy to remember maneuvers; if you must do a spellcaster, give him a few simple spells that he always prepares, instead of making the player pick the NPC's spells from a huge list every day.
Sometimes I feel like giving the player an NPC party member is "cheating," since it's not really a solo campaign then, but so far my player has really enjoyed the NPCs, both in their personalities and in how much they help her survive in combat. And they aren't
always a benefit: if the battle suddenly goes south, the PC can't just turn tail and run, using a high stealth check to escape, if he has an unconscious or unstealthy party member slowing him down.
Basically, a solo campaign is a perfect opportunity to focus on the parts of an RPG that recede into the background with a larger party (thematic development, narrative character growth and story hooks), and it doesn't have to be boring because there are so many ways to still have exciting combat. I hope these suggestions help!
