I've been playing guitar for a number of years now and I've checked out alot of info about finger excersises. It seems to me that when you're working with rod puppets pulling and pushing switches all the time you actually need your fingers. But when you're just performing a handpuppet I agree that arm, shoulder and back strength are most important.
I only know about finger excersises. And perhaps a stressball might be ok. The danger is though that the more you actually try to train your fingers on strength the more cramped up you'll get. Like someone said earlier flexibility is much more important. Training your fingers strength will only put more pressure on the finger joints. I haven't heard anyone talk about bloodflow though. Seems to me that when you have your arm up almost all day all the blood stops flowing to your hand.