This is a very old thread.
Most of the muscles that control your hand are in your forearm, not in your hand. The tendons pass through your wrist. (Put your right hand around your left forearm and feel the muscle move as you move the fingers of your left hand.) If the issue is muscular, icing along the muscle and tendon can help.
There are also a number of small bones at the junction of your hand and wrist; the carpals. It's possible for these to get out of alignment with each other, moreso given that the deadlift basically pulls your joints apart.
If your issue is because a bone is out of alignment or you damaged one of those bones, icing won't do much for it. Either let it heal and/or get things looked at by a professional.
In both cases, you're going to have to judge whether what you're doing is making things worse, whether you can live with it, and how much time you're willing to let it heal. Rest time, while it may heal things, may not actually "fix" whatever underlying issue caused the problem in the first place.