Adventure games as others have suggested can usually be played with a mouse, ie, point-and-click adventures like the Broken Sword and Monkey Island series. They're old but still good.
For RPGs, you can play Neverwinter Nights and Dungeon Siege completely with a mouse (I know, because that's how I prefer to play them),and of course there are other action RPGs such as Sacred, Divine Divinity, etc.
As already suggested, most RTS games can be played with a mouse as well, so don't forget Age Of Empires series, Age Of Mythology, Rise Of Nations, Cossacks, Warcraft and others.
On a side note you can technically play some older FPS games like Wolfenstein 3D, Doom, Heretic, Hexen etc., with one hand as well, because the default settings allow you to move your character through the levels with the mouse, and with a multi-button mouse you could bind other functions to the mouse instead of the keyboard (jumping, strafing etc.). Trickier to play that way, but possible.
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