The white rubber part is a locking plug that can be inserted in between the arm ends of the quick release buckle. Having the rubber plug inserted prevents any accidental release of the quick release buckle, and also tightens up the fit. The looped end of the plug can be attached to the thumbscrew to keep it secure when not locked into place.
If you do not have a quick-release anti-vibration locking plug, you can get one by purchasing the Grab Bag of Mounts.
Quick-release anti-vibration locking plug inserted between the arms of the
quick-release buckle and the adhesive mount.