Go back to basics.
Download MAME and let him play some of the classic arcade games. Most would need at least a fire button (still easy to map to a single-hand controller), but there are also a few great classic games like Frogger and PacMan/Ms PacMan that only require a joystick. They are also really simple to grasp while maintaining a good challenge.
I also assume he's at least 40+, so he should be already familiar with this type of arcade games from his youth.

Sorry for your uncle, man. I just imagine to pass all your life getting knowledge and loose everything with a finger snap. That sucks...
I would suggest Tetris.

I hope he will get better soon. I don't know too much about strokes, and i don't know if this can be cured, but i hope so.
Have a nice day.

People already mentioned Osmos, EDGE, Night Sky and Badland (I concur), so I'll just add Solar 2 to that list. It's similar to Osmos in concept, but instead of clicking in the opposite direction of where you want to go, you just click exactly where you want to go. There is a goal, but you can also just float in space and watch other systems.
I hope your uncle will recover soon. All the best for him and you.

When he recovers a little bit you might want to consider getting him a one handed controller. Best of luck to your uncle. :)

Sorry about your uncle. I have an aunt that had a couple of strokes, so I kind of understand how traumatic that can be. Here's hoping for a steady recovery.
I don't believe anyone has mentioned Windosill yet. It's in the latest Humble Weekly bundle, along with some other games that he might enjoy after he's recovered a bit. Windosill is quite simple, requires only a mouse to play it, and isn't bogged down with text. Though it is really short.

If I can think of anything else, I will post it here. ^^

Edit: Cut The Rope looks like something he'd enjoy. It can also be played one-handed, and isn't too complex at all. Plus it's only five bucks on Steam right now!
And Aether is also a great one. The browser-based version I linked to is 100% free, though you can also get an improved version of it in The Basement Collection. Either one is good. Though that would probably be better for later, once he's gained a bit more use of his other hand.

Don't Starve can be played with just the mouse

Just thought of Limbo. It's an excellent and easy game, although it gets harder later on in the game. He can always play the game until it's too hard, and move on to another game.
I hope he's getting better. :)

Just thought of Limbo. It's a nice, easy game, although it gets harder later on. He can always play it, and when it gets too hard, move on to another game.

Super Hexagon

bejewleded!

FTL, Prison Architect, Dwarfs?!, Hero Academy, Swords and Soldiers HD, Jolly Rover, James Town, Poker night at the Inventory, Magic, World of Goo Are all games that could be played using one hand. Not sure most of these will fit though.

Try those turn-based strategy games like Civ 5

I think you can also try some turn based-rpg. From what you say, he seems to like story, so something like Final Fantasy could work just fine.

Try Hammerfight, you can control your ship with your mouse, so you only need one hand.

I'm terribly sorry to hear about what happened to your uncle.

As for suggestions...I think that apps are more "single-handed friendly" and if you don't have the funds to buy him an expensive tablet, then I would suggest downloading bluestacks. It's a program that lets you run apps on the computer so long as your pc is good enough.

I also think you should give board games a shot with him. Something like checkers or even connect 4. Just remember to go easy, wouldn't want him to further harm himself by getting frustrated. :/

There are much Games for Android or IOS: Angry Birds, Where is my water?, Cut the rope, Granny Smith or Rayman Jungle Run just to name a few. All of them can be played with only one Finger. Maybe a Wii could be a good idea, especially Wii Sports with Tennis and Bowling can be played by only shaking the Wiimote. Or Moorhuhn (Crazy Chicken) is a simple Browsergame where you only need a mouse http://www.bella-ben.de/shop/moorhuhn_com/?nid=39 .Good luck for your Uncle.

World of Goo, Puzzle quest (there is a new one, free to play, not bad), Gyromancer(i dont like it as much)

Audio Surf, Cut the rope/obulis, Dungeons of Dredmor (mods), Dungeon Hearts, Droplitz, Eufloria (its like rts-lite, very very nice to play though), "Typing of the dead: overkill" it has "House of the dead: overkill" and its on-rails shooter. Might work.

FTL: Faster than light, Knights of pen and paper, Little inferno, Poker night 1/2, Quantz, Reus, Revenge of the titans, RUSH, Toki Tori, TOME, Tidalis,

TRIPLE TOWN! its free on facebook, has a nice/nonsocial version on steam.

Yosumin!, Zen Bound 2, Waveform,

Chainsaw Warrior/Space Hulk/Civ 5,

Eets, Eets munchies,

Fallout/2/tactics, Full Spectrum Warrior/Knights of the old republic/2(not mmo one),

Mousecraft?, Osmos,

Serious Sam: The Random Encounter,

Xcom:EU

DLC Quest

Splice

Home

Divekick (Only uses 2 buttons :) )

Sounds like you want mobile games

Why the long story about your uncle, couldn't you just ask for a simple and easy game ?...

Point & click games like The Walking Dead.

scriblenauts is pretty fun, puzzle game for well kids. I enjoyed it quite a bit =)

I think Tiny Thief fits the bill in this case. All mouse controls, no dialogue (written or verbal). Not sure if the logic required for some of the levels is within his grasp currently but it might be worth the time to check it out.

Fallout 1 and 2 don't need a keyboard. Game Dev Tycoon (tycoon games usually don't need more than a mouse) or any of the Civilization games.