What are some good exercises to build muscle for someone with arthritis?

Start with walking. Brisk walking is great exercise, low impact and with minimal joint strain, and can be increased incrementally to provide maximum benefit. If the arthritis interferes with walking, and if you have someplace to go to do it, then swimming is just as good or better, and even less traumatic to the joints than walking.
Exercise. My favorite is pool therapy or a water arthritis class. You are non-weight bearing so you can buildup muscle without taxing your joints.