Classic. Some people slip an O-ring over the end of the button, some people wedge something between the handle and the seat, I think I remember some people putting a small screw through the end of the button just to add some weight and damp it...
Mine use to do that, but put my knock-off yeti cup there (wedged between brake and seat) and it no longer rattles....also holds my cup well sorta..well unless I brake or accelerate