Fate/stay Night (the PSP version).

I would try out Zack & Wiki: Quest for Barbaros' Treasure  for the wii its a puzzle solving one handed game

Temple Run will work. It is for Windows Phone. You will have to tilt the phone, though.

Davideo7 : I play the Shining Force games in the retro games room on Sega Genesis with one had all the time, they are some of my kick back and relax games...

Davideo7 : Shin Megami Tensei IV is pretty easy to play with only the left hand

legend of zelda: spirit tracks

You can complete the whole entire main story of Bravely Default with just one hand. And, I highly recommend this game, because it's got to be one of the best RPG's out there. By the way, I hope your hand gets better soon, David.

Dig Dug! It is a great game, and doesn't require 2 hands.

Davideo7 : I play every game I can with one hand since my left hand/arm is paralyzed.
I use a multi-button mouse (Logitech G602 series) and it pretty much has all the buttons I need. There are a few games that I wish I had a many more button options for, but the price of those mice as well as their quality is truly lacking.
I cant play many games for systems like the N64 and the PSX stuff do to their buttons being too numerous in play though.
Also the metroidvania type of games are really tough to play do to the maneuverability they require. A D-Pad is tough to simulate while using only one hand. I often make the first four buttons on my side of mouse into my Up, Down, Left, Right setup. It's not perfect, but it works.

This is why I only play games on Vizzed. Mainly the RPG games. I'm hoping to eventually get enough Viz to by the PSX playability option so I can play even more RPG games with those systems. But for now, I just play the FF series over and over again. It's better than nothing when you are stuck in bed 24/7 and only have use of one hand.

I'm truly grateful for Vizzed. I might have gone entirely insane without it.

Davideo7 : Hi there! If you have a wii and use wireless internet, you could download the free trials of Bit trip Beat, Flux and Runner. They are neo-retro games on both wii ware, and on their own disc all together in Bit Trip Complete that are one handed. When I say neo-retro, I mean that they really do combine retro and modern graphics, which is very cool! They are also highly musical. Everything you do in the game produces another beat, and when you do the game perfectly, it sounds sooooo beautiful!

You like point and click games? The Grim Fandango remastered for the PS Vita would be a great one handed game because you can use the Vita's touch screen to play. From what I have heard it is a really great game, never played it myself so I can't say one way or another. If you have a Vita and decide to download the game from the ps store let me know what you think of it.

Art Style: Cubelo uses one hand, I'm pretty sure. There's also Mario Super Sluggers where they give you the option to play without the nunchuk. I think Echochrome only uses the camera and a button to speed up time. OH and most games in Sportsfriends can be played with just the left or right side of a controller, if you want to play a local multiplayer game. And two people can use one controller to use their own characters. Also, if you have a kinect (god help you) I've heard Child of Eden is one of the few actually good kinect games, and the only interaction your left hand would have is bringing your hands together briefly to change weapons, or raising your hands up to use a bomb.

This is a controller made for one-handed players, so that they may play any typically two-handed controller game.  (All you'll need is a Playstation controller USB adapter for your PC...)

... and, it's right handed! 

http://www.ebay.com/itm/Playstation-ASCII-Grip-One-Handed-Controller-FREE-SHIPPING-/301513189140?pt=LH_DefaultDomain_0&hash=item4633962714

http://www.blood-is-red.de/mw/images/Asciigrip.jpg

On Vizzed, this includes Playstation, SNES, NES, all the Gameboys, Genesis, MAME, Jaguar... etc. games.

I realize the disturbing amount I'm owed, now, (considering how silly it would be to list out every Viz game now available to your right hand, because of this post,) but I'm willing to settle for 250,000 Viz.


Get Well Soon!

-MAP

Davideo7: If you like rhythm games, there's Beat the Beat Rhythm Paradise for Wii. The American version is called differently, something with Rhythm Fever, I think. Anyway, personally I think it's a great game. It only uses the A and B buttons, I think. So that's perfect for using only one hand. There's also a ds version of the game, but since I don't own that one, I can't really judge it.



Legend of Zelda : Phantom Hourglass (possibly the others too but I don't have those) you have to use the stylus for just about everything, so maybe if you place the ds on your lap or against your knee, you won't have to use your left hand to hold it.



Magical Starsign works pretty much the same; using the stylus, I think.  In my opinion it's a nice RPG.



Wario Ware Touched



Flow on PS3

Davideo7 : I know for a definite fact that you can play Shin Megami Tensei IV with one hand if you use the 3DS stylus. I do this constantly because my hands hurt.  It's the same with Shovel Knight, Phoenix Wright, and Bravely Default. All of these are for the 3DS or can be played on the 3DS. 

I really, REALLY recommend SMT IV and Bravely Default most of all for one-handed play. Even if you have super-tiny hands like I do, they're fairly easy to use one-handed with the stylus, and they're fun.

Try Hotel Dusk

Davideo7 : ezrpg on smart phones is really fun it plays itself for the most part and it's totally free and I haven't noticed many ads on it they even give the pay to win in game money out fairly often I've had 2000 on my main toon

Scribblenauts Remix (Moblie), Rhythm Heaven (DS), Animal Crossing New Leaf and Tomodachi Life (3DS), and I guess any Wii game where you just use the Wiimote.

Davideo7 : these games aren't exactly great if you have played them before, but I'll list them off anyways:

Mother/Earthbound series (RGR NES/SNES/GBA)

Ace Attorney series:

   -The original trilogy (RGR GBA (Japanese), DS, 3DS, iOS)
   -Apollo Justice (DS)
   -Investigations - Miles Edgeworth (DS)
   -Investigations - Miles Edgeworth 2 (DS (Japanese/Translation hack))
   -Dual destinies (3DS)
   -Professor Layton vs. Ace attorney (3DS)

Professor Layton series (sadly, I have never played a Professor Layton game, but I've heard they are quite similiar to the Ace Attorney games) (DS, 3DS)

Even though some of these games require pressing buttons on the other side of the controller, you can take your time as these games aren't exactly action based (they are turn based games and puzzle games, respectively).