Sometimes I wanna run a game.
Sometimes I have a gaming group with schedules that conflict heavily.
I guess the easiest way would be to make do with what I have. So here's some tweaks that I may put into action sooner or later.
These are just random considerations.
Healing Surges. The Healing Surge option from the DMG looks like it'd be a good fit. In-combat self-healing would definitely help in increasing the lifespan of a solo character. Personally I'd go for the superheroic option and allow it to be used as a bonus action. Additionally I'd allow it to be used to automatically succeed on a saving throw a character failed.
Cleave. Definitely something worth grabbing from the DMG, as it will help classes like the Rogue keep up with multitudes of enemies.
Conscious Death Saving Throws. Normally a character reduced to 0 HP falls unconscious. As a solo character would not be able to receive any help or healing when fallen (unless they had NPC allies tagging along), they instead retain consciousness and are allowed to take actions, even when stabilized. Death saves, instant death, and all associated mechanics still function normally. Potentially, stabilization whether through natural means or magical will incur a level of exhaustion.
Echoes. Considering there's a whole group of players, that's up to four times the GMing for one game world. Taking a cue from the Souls series, why not allow a bit of interaction between them? Any area where one PC left off would leave them as an Echo, essentially a friendly NPC of that character that can be summoned. They're functionally identical to the player character in question, only they vanish if they reach 0 HP. Events caused by one player could potentially ripple outwards into other players' game states as well- If one player caused two monster tribes to band together, they could potentially band together in other players' games. A global 'World Tendency' a la Demon's Souls might also work for this kind of thing.