Can young children have essential tremors?

My grandson frequently experiences uncontrollable shaky hands (or hand tremors) especially right after he wakes up, but also through out the day. He is 5 and has experienced this as far back as 1 years old. His pediatrician said he would grow out of this. He didn't think it was anything to worry about. Honestly, this issue does not bother my grandson one bit and he seems to be able to do anything that any other kid can do. It's just disturbing to watch his hands shake so much when he lifts a glass of of water to drink. Again, this does not happen all of the time, but it does happen regularly every day. Does anyone have any thoughts or suggestions? My son is going to take him to a different doctor for a second opinion. What kind of specialist should he see?