Holding a 2 handled tray with both hands is going to give you the most stable footage. You really have to concentrate on holding it steady, panning smoothly, etc so maintaining good bouyancy and trim needs to be second nature. There is no IS in the camera so every little movement shows up, you can stabilise it to a certain degree in post processing. Yes having more mass (inertia) is going to also help steady it a little but you'd have to make it many times more massive, then neutralise that negative buoyancy, to make much of a difference I feel. I would be interested to see the results of such an experiment.
Here is my first gopro video (and first time holding a video camera underwater) so it could be smoother. This was with a 2 handled tray... mounting to the head, chest or any other body part makes for unwatchable footage IMO.
[video=youtube;93J7a3Srf1Q]https://www.youtube.com/watch?v=93J7a3Srf1Q[/video]