If you have arthritis in your thumb, is it a good idea to exercise your thumb?

May be. Such arthritis needs to be evaluated by a rheumatologist.
Arthrtitis. Exercise for arthritis is designed to keep the joint moving, but not to put toomuch stress on the joint. Stregtheningthe muscles around a joitn without joint irritation also will help stabilize the joint and reduce joint force especially in extrems of motion. So yes but not too much that it hurts.