Tic Tac Toe, stuff like that? There are tons of these things. :)

Super hexagon.

Little inferno is a great game that only requires the mouse to play. it is a puzzle game so it could be hard to progress, but it is just about burning stuff so you don't really need to progress all that fast :D

Instead of video games, what about some crossword puzzles? They would (probably) also help with training his knowledge.

That reminds me ... what about You Don’t Know Jack?

Edit: Also most point&click adventures should work fine with just one hand, though some of them have tough puzzles.

Sorry to hear that. Good to hear he's making a recovery.

Anyways, I suggest you browse casual games on Steam. Games like Clickr might do the trick for now.

I'm very sorry to hear about your uncle.

I think, maybe just some casual games? Solitaire and the like.

good news Again I'm not spreading fake shit. There is a way to make a controller with tongue and butt movements. There are videos to show how to make them work for you.

try turn based strategy games as they would wait for him to make his input without rushing him.

im sorry to hear that :/.
maybe its not that simple, but.. http://www.youtube.com/watch?v=K5_ZkHj5RtE

I'd try some "contemplative" games like Dear Esther, when mostly all you do is walk. But if he can only uses one hand I dunno if he will be able to play well. Maybe you could play it for him to watch.
I remember playing Pokemon back then with only one hand when I broke my left arm, but if you say he cannot read very well at this point..
My condolences about whatever happened to him :( It's great for you to trying to cheer him up! ^^

Chess sounds great

sorry to hear about that..

well, i think Civ 5 can be played with just the mouse (one hand).

then there's Osmos which is a simple game

Plants VS Zombies

King`s Bounty series

I cant remember, can play this game with one hand or not, but FTL Faster Than Light

I would suggest Hearthstone. I mean, depending on how difficult reading is for him it might be too reading intensive. But as card games go it's pretty simple and is completely played with the mouse.

Hope you get some great suggestions!

Ehm... Crayon Physics? I played this game and it was quite simple but... I enjoyed it like a child! I hope he gets over it soon!

AUDIOSURF! :D
easily playable with 1 hand and it's great for improving coordination :)

Hmmm, probably the old Monkey Island games.

visual novels i think.. Point and click?...Don't starve maybe?... Tetris is great too you have to use a brian little by little. "Kick it up" for pc? it helps with timing etc... And dunno others.. But search for point and click games, or easy games visual novels etc..

a stroke of what? luck? genius?

I'll be talking about steam games

BIT.TRIP series might be good... maybe...

Fortix?

Chime!!!!

I would say mcpixel but... that will probably end up hurting him more...

Snuggle truck

Thomas was alone

World of goo might be a fun game too, maybe in "Free mode"

From that list I'll make a top 3

1) Chime

2) Snuggle Truck

3) World of goo

In any case hope any of those helps you :) good luck and best wishes

http://store.steampowered.com/app/27020

I think it's only 10-15 minutes long and i believe you can play it with mouse only if you rebind controls

Really sorry to hear that...
What you think about Back to the future? I hope that it is not too complicated.
On the other side you can try Fortix 1&2, Limbo/Pid, Faerie Solitare, Little inferno or some pinball games.

How about you let him watch you to play some games he can't play now? The only game that I can think of is Audiosurf.

You could try point and click games, I know most people hate them, but I'm obsessed with them, they just have such a good story, I saw a few alright ones on steam but don't remember there name. I would probably try suggesting something similar to mobile games, like something such as angry birds where you can simply fling birds at pigs is one handed and simple enough of a premise. They have it on PC and there could be lots of others that was also ported over.
I hope everything goes well for you, similar thing happened to my dad so I hope things get better for your uncle, especially if he starts playing games.

Buy him an iPad or something. Lots of simpler, shorter one-touch games.