Just received my second GP Hero 3 in the mail yesterday. A replacement for the original which is now an anchor at the bottom.
So this time I decided i'm not doing anything without a tether. I've ordered some very cool tethers from Big Kahuna Camera Mounts http://bigkahunamounts.com/
Until those arrive, here is the DIY temporary tether.
Bought an 18", 45lb test leader line and a simple carabiner. Connected the clasp of the leader line around the rod in the door hinge. Attached the 'biner to the other end. it'll connect to something - say a strap on my PFD. Or could put a velcro loop on it too. Total cost: $4.