How are military jets given official names?
Typically for modern US aircraft, since the military pays the contractor for the aircraft's development, so they are paying for it, and get to name it. This article says the USAF named the F-35 as the "Lightning II". However, the reality is much more likely, the F-35 Joint Program Office (the JPO), named the aircraft, given it international status and being a joint aircraft, and the announcement was made by the USAF. The JPO, being a joint office, always has a Program Executive Officer (PEO), and a Deputy, from both the USAF and USN (alternating positions). The F-22 was likely named by the USAF F-22 project office, which was based at Wright-Patterson Air Force Base, Dayton, Ohio. See here.

However, this isn't always the case, at least historically. For example, the Lockheed Orion &amp; Hercules,  (and the Constellation) follow a tradition of Lockheed of naming their aircraft after celestial objects, stated here and here. The A-12 was also named by flight crews as the Cygnus (swan), another constellation, to follow this pattern. So, some US military aircraft, in the past, have been named by the manufacturer.

As another example, the F-111 is believed to be the only US aircraft to go through its service career without being officially named. On its retirement, the USAF designated it the "Aardvark". While the name was made official by the USAF, it came from the term the crews used for it.


  Its American crew called the F-111 the Aardvark. It was a good name. Like the South African aardvark, the F-111 was a solitary night hunter. Like the aardvark, it rooted in the dirt, guided by excellent senses. And like the aardvark, it had very long range. For much the same reasons, Australians called it the Pig.


So, while in the past, sometimes the manufacturer named the aircraft, sometimes the crews, it would seem, today, the US government, through their military force, chose the name.