Why would part of my right hand falls asleep at night?

Nerve problem. You have pressure on a nerve in your arm. Either the median nerve in the carpal tunnel at the wrist or the ulnar nerve at the cubital tunnel at the elbow. Carpal tunnel syndrome will include the thumb, index and long finger as cubital tunnel will be the little finger. Stop sleeping with your wrist or elbow bent (flexed). Then see s specialist.