As Mike wise pointed out it is not an issue with the FAA unless you do something else during the installation that requires a 337.
Please consider placement and behavior during operation. Don't install it in a manner that will cause damage or jeopardize flight controls. I've seen one mounted to a flap bracket on a Piper not sure how they thought that always a good idea. People seem to engage in activities that they otherwise wouldn't once they are being filmed. Don't do anything dumb, like buzz a house or roll a C172.
I'm a fan of zip ties, they've worked well up to 200mph and +6g. I've used the suction cup and it has worked well to 180mph and +8g.
Also if you're a renter check the policy on cameras. My local school and the other school in the area have policies regarding GoPros.
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