I knew someone would say that XD

Settlers 2

Someone made a gog mix:

https://www.gog.com/mix/best_played_with_mouse_in_one_hand_beer_in_the_other

It's a bit out of date, but still useful.

I recommend Reigns. It's actually even better with painkillers and alcohol.

Kingdom New Lands I believe you only need the direction keys to play. Great game too.

I hope he heals soon.

Anyway, most turn based and RTS games fall into this category or RPGs with a top-down perspective - though it may take some time searching the screen for the right buttons/icons.

Games I'd recommend (all on GOG):
- Grim Dawn
- Titan Quest
- Disciples 1 & 2
- Heroes of Might and Magic games
- D&D Dragonshard
- Battle Realms
- Spellforce 1-3 (the view will need to be adjusted, but it's easy to do)
- Torchlight 1 & 2
- Pillars of Eternity 1 & 2
- Neverwinter Nights 1 (2 as well if you adjust the view)
- Buldar's Gate and Icewind Dale games
- Darkest Dungeon
- Banner Saga 1-3
- Etherlords 1 & 2
- Shadowrun games
- The Longest Journey
- The Incredibile Adventures of Van Hellsing - FInal Cut
...
I can't think of any more at the moment.

still in need of games?
rusky lake series

Hidden Object games can be played with one hand. You can even find some on GOG.
And PC board games, like Monopoly, Cluedo, Backgammon, Scrabble.
And Break-Out games.

One Finger Death Punch

If you have a mouse with more buttons, you might be able to play some ARPGs

Get rid of Monopoly, and I'll accept this post.

I play Monopoly all the time on my computer.

Okay, bad example.

Old consoles have 3rd party one-hand controllers.
Modern consoles have too many buttons, joysticks, accelerometers, making one-hand controllers unusable,

Yeah, Monopoly isn't a game.

How is Monopoly not a game?
I frequently play that on my PC.

The first two Fallout games can be played with one hand, no need for quick moving (or thinking) with turn based.