Drones must be registered with the FAA starting December 21, 2015. The cost is $5, but it is waived for the first 30 days so a Christmas present this year will cost nothing to register. The FAA's press release says that the owner will need to register, so it sounds like the recipient will need to take care of that.
From the FAA:
"Registration is a statutory requirement that applies to all aircraft. Under this rule, any owner of a small UAS who has previously operated an unmanned aircraft exclusively as a model aircraft prior to December 21, 2015, must register no later than February 19, 2016. Owners of any other UAS purchased for use as a model aircraft after December 21, 2015 must register before the first flight outdoors. Owners may use either the paper-based process or the new streamlined, web-based system. Owners using the new streamlined web-based system must be at least 13 years old to register."