The autoharp requires very minimal (and non-strenuous) use of the left hand.

The psaltery doesn't require any left-hand finger action at all -- you just hold it in your left hand and the right hand does all the work. Likewise for the Jew's Harp, the glockenspiel, the melodica, the french horn, and the spoons.

Tambourine, maroccas, and other percussion instruments of that sort are extremely easy to get started with, and can be lots of fun.