Oh my this is sooo freaky, I was just looking into this because I have also notice in the last few months my 3 yr olds hands shaking in the early am, holding a cup of water, or any 'fine motor' type of activities. Only difference is there is a long line of genetic essential tremors on my husbands side of the family. His sister has them, his mom, his grandfather, then also his grandfathers sister. So bottom line I'm pretty sure my son also has the same essential tremors.
My MIL said she didn't notice hers until she was closer to 8 yrs old, my SIL can't remember, but did state hers is exasperated by low blood sugar, caffeine, and fatigue. Even so once I can get the shaking on camera, I will be making an appointment to have his blood sugar tested (as type 1 diabetes runs on my side of the family) and to also rule out any other type of reason to cause the shaking. My suggestion would be to do the same.