Any turn based game, like Valkyria Chronicles. Japanese are really good at making one handed games. Yeah, I'd go with turn based JRPG, simulations, strategy etc.
If you can afford aftermarket controllers, there are onehanded controllers out there, any Guncon 3 game like Time Crisis 4, any PS Move game like GoldenEye 007:Reloaded, any race games compatible with a wheel (some maybe possible with sixaxis), some arcade fighting games with arcade stick, DDR with the controller mat thing.
But I'd go with the JRPGs. Enough fan service going on, you won't need two screens.