Is there any size of tethered kite that is regulated by the FAA?
As one of my instructors was fond of pointing out, there's pretty much a regulation for everything -- in the US kites and balloons would be covered under FAR part 101 which is pretty short and to the point as far as government regulations go.

The big rules regarding kites/balloons kick in for a kite "over 5 pounds" or a balloon "over 6 feet in diameter / 115 cubic feet of gas capacity", and are:


You have to keep them at least 500 feet below any clouds.
You have to keep them less than 500 feet above the ground.
You must be at least 5 miles from the boundary of any airport.


If your kite/balloon is going to be more than 150 feet above the surface you're also supposed to coordinate with the closest ATC facility (who would theoretically put out a NOTAM saying "John Doe is going to be flying a kite/balloon/thingy here, between these hours. Please try not to hit it.")

There are also some lighting requirements (if you fly the balloon/kite at night), and visibility requirements (but really, who wants to fly a kite in less than 3 miles visibility? That seems like pretty lousy kite weather.)



As a favor to your local pilot community (and to prevent having your wayward kite dragged off by a Cessna on short final) it's probably best to observe the three "big rules" above regardless of the size of your kite - a 6-ounce piece of balsa and cellophane can still make a lot of people's days more interesting if it appears in an unexpected location.