A brief of different editions of D&D for solo

AD&D 1e is the best edition for solitaire play. It not only comes with a section on "Random dungeon generation for solo play", but with the Random Wilderness Generation table, it can also be used for solo wilderness exploration, and the many, many other charts can easily be adapted into random generators as well. This means you can randomly create whole towns, npcs in detail, encounters, and even new monsters and magic items. Using all this content generation with the random encounter chances, and the solo player can readily simulate the procedural creation and exploration of a roguelike videogame, such as the Diablo series.

AD&D 1e is a very rules-heavy game, and has many subsystems governing player actions, and many rules for resolving those actions. This makes adjudicating certain situations just a matter of following the guidance in the book, and whole games could be played without any external resources. Although, AD&D does place a lot of importance on the role of the Dungeon Master, but those situations that the book does not cover can be filled in with a GM emulator, such as the Mythic GME.  This, however, means that playing AD&D solo is excruciatingly slow, and requires a lot of bookkeeping.

D&D BECMI is very similar to AD&D but is much simplified. In fact, many of the rules and systems described in BECMI have a direct counterpart in AD&D, but with simpler mechanics.  What the BECMI rulebooks are missing, though, is a random method for creating a map or a dungeon, but they do suggest a procedural method for creating one, and then randomly stocking them with monsters, treasures and traps in the case of dungeons, or with random encounters in the wilderness. The Expert book also comes with maps of Threshold, Karameikos and the Known World, so maybe creating a custom map is unnecessary when the player can follow the adventuring rules to gallivant across the prebuilt setting.  The looser nature of the D&D rules means that the player is freer to pursue any imagined course of action, without consulting the rules, and without needing a die roll for everything.

5th Edition D&D is built upon the chassis of the d20 system, and as such has a universal system of conflict resolution, which the AD&D and OD&D systems lack. 5e's basic method for handling any situation is to assign it a "Challenge Rating", and then resolve success based on the roll of a d20, modified by ability and skill scores.  This makes the base system much easier to learn and run, and provides an elegant solution to any unknown situation. The 5e DMG also took cues from the 1e AD&D DMG, and contains charts and tables for random generation and stocking of content. The dungeon generation chart in particular is more balanced than its 1e counterpart. 5e is more story oriented than 1e, though, and as a result has a section devoted to generation of quests, NPC motivations, and plot twists. This could theoretically elevate the nature of the game from the procedural dungeon crawling focus of the earlier games to one more goal oriented, but I haven't actually tested it out.

5e is also an evolution upon 3e D&D, but is a lot lighter on the rules. An explicit design choice in 3e was to reduce the role of the DM from the adjudicator of all scenarios to a referee of the rules, and there were rules for many, many scenarios. The rules heavy nature of 3e makes it more workable to facilitate a game solo as there are a lot less unknowns. 3.5e and Pathfinder are still popular choices when it comes to tabletop roleplaying, but I've never played them.

Rolling for initiative

I don't like the convention of having the players roll for initiative at the start of combat. It's an invention brought in by the 3rd edition of D&D and maintained all the way up to the present day. The problems are manifold and have been belabored to death elsewhere, so I'll just point out my solution. AD&D and BECMI don't work like this. In both systems, players must make an intention of action BEFORE the initiative die are rolled. This sounds like a wargaming practice, but it also works great in narrative combat.

Focusing on D&D narratively, as if we were playing purely in 'theater of the mind', the players would be doing whatever they want, until one of them comes into a situation and will say "I attack".  At that point, their attack should automatically succeed, unless it is opposed by an enemy or monster, who decides to fight back. The monster should also announce its action, such as "I breathe fire." Only then are initiative die rolled, simply to determine which action goes first. After that single round of combat, narrative play would resume, unless the two combatants opt to continue a series of attacks (or opposed actions)

This is a subtle genius woven into the original D&D framework that is not effectively communicated in either the rulebooks, or from most groups online. I've been part of AD&D and Original D&D groups where the DM used 3e style individual initiative, simply because it seemed to be more fair in giving all players a turn. For that matter, I found the O/AD&D  turn sequence to be much more fair, as every player has his or her own say every turn if they have to describe their actions to a caller, who would then relay those actions to the DM. I personally don't use the caller, or rather as a DM I also act as the caller, and ask my players to describe all their actions per turn to me first before I resolve their actions all at the same time. I describe my method more in depth here

I read a blogpost on AngryGM where he disparages the initiative roll as the "whoosh" of loading the combat screen from a videogame rpg, and I agree. But using O/AD&D style encounter rules, and rolling the initiative die only IF two sides decide to take opposing actions, and only AFTER they've made their declaration, allows you to weave combat into narrative gameplay without breaking the flow of the narration, and still allows players who do not wish to be in combat to go about and do their own thing. The only thing to remember, to keep everything balanced, is that 10 rounds of combat equal 1 turn, so you can pace the players' actions accordingly.