My understanding is that hands will move unsteadily if the body IS stiff. There is a lot of motion that must move throught the body of the rider, like a wave on energy moves through water or, like the way you might swing a towel like a whip, to make it "crack".
A rider must absorb the energy in their body and let it come "through" and out of them somewhere. This should be their hip/pelvis. If they are stiff there, then the energy travels further through the body and comes out the hands, making them move up and down. Instead, absorb the energy in the hips AND the elbows. The elbows must be loose enought to move in order for the hands to stay still.
Another thing that some people do is work on training that stillness into the hands by holding a string, tied to the D ring of the saddle, which is about 25cm long or so, while trotting. This helps the rider keep their hand in one place.