Team Fortress 2. Play a Pyro.
The Aurora Engine games, KOTOR and NWN and such, would be.. doable with just a mouse? But having sunk many thousands of hours into NWN, you really want keyboard control; the click to move setup isn't particularly precise. Possible, but I'd honestly caution away from them, the mouse control is that bad for moving.
RTS style games, things like Warcraft, Starcraft, or Dawn of War would be at least playable with just a mouse, even if they're at their peak with hotkeys. I don't like doing product recommendations for things I don't own, but maybe show him the Razr Naga, or one of the other various gaming mice out there? They come loaded with a whole mess of thumb buttons, such that you could very well replace most hotkeys that a lot of games would need. MOBA style games, RTS, something like a Naga would open up a vast amount of possibilities - maybe even certain shooters, depending on the individual game in question.
Shadowrun Returns is built in the same engine that XCOM is, or at least, an absurdly close version of it, such that it also would function well enough in one handed playing. As would the version of Chrono Trigger available for Android (Don't have an iOS device to check), which is easily handled with just a thumb. Dragon Age: Origins would be something to look into as well, for the same reason as the Infinity and Aurora Engine games; you pause so often in combat that you have plenty of time to control the game with a mouse.
I'd offer up World of Warcraft, also. He may not be able to get into raiding very easily (though, again, one of those keypad mice..), but there are a fair number of MMOs that are slow enough as to allow mouse control. FFXIV is one of them, it's built with a 2.5s GCD, with the effect (and intent) of slowing down combat and making each cooldown a tactical decision. EVE Online is well known and derided for its spreadsheet gameplay, but it's still mostly mouse driven - I haven't played it in like five years though, so I can't offer a realistic estimation as to playability, and even then only in the relatively low tier of gameplay I got into (Once I got my Dramiel the rest of the game was pointless), but I wouldn't be surprised if it was one of the better options.
Visual Novel type games are out there, those generally require no complex interactions. Admittedly most of them are of a.. less than wholesome nature, but still.
Minecraft, maybe. Depends on number of thumb buttons. 2-3 should suffice.
Most of the older, 8-Bit RPG type games are generally set up to emulate on a PC using just the arrow keys and Enter or Shift or some such, so those would be a good choice. I can't recall if FF8's PC port can do it, but it wouldn't surprise me an awful lot. By that token, Audiosurf would be a thing too, since it's just using arrow keys.
If I can think of any more I'll post again. But dredging through my Steam library isn't sparking many thoughts. Hearthstone, maybe?
