I've been playing PC games one handed lately (nursing an infant with the other) and I've been choosing games that suit. I've got a mouse with the following buttons/actions: left, middle, right, thumb1, thumb2, wheelUp, wheelDown.
If you're working with your non-dominant hand then innately mouse-only turn based strategy, adventure or management games are going to be your friend. At least while you retrain your mouse usage. Things like Civilization, Hero Academy, SimCity and Monkey Island.
I've also found action RPG games with a small amount of keyboard commands are extremely playable if you remap the keys to your mouse's spare commands. I've been playing Diablo 3 with all six skills and health potions mapped to the mouse. It works really well and I only feel disadvantaged when I want to shift click to stand still and attack - which I'm one button shy of being able to map. With good character and skiill choice you could probably offload a 5-minute buff to the keyboard (numpad-enter or tab).
I haven't figured out how to play FPS games competitively yet, but binding move forward and one side-strafe to the mouse might get you started.