The F-15 was formally named the ‘Eagle’ in the lead up to its 1976 introduction into operational service. I would imagine, no evidence – just conjecture, that the name would have stemmed from the churning of nationalism in the lead up to the USA’s Bicentennial. In such an environment naming a new, impressive fighter after the national animal symbol, the Bald Eagle, would have been natural.
Source: http://www.faqs.org/docs/air/avf15_1.html
The F-16 received its name courtesy of a name-the-plane competition being won by Tech Sgt Joseph Kurdell.
Source: http://www.f-16.net/articles_article10.html
The F-22 received its name as a suggestion by the then USAF CoS’s wife (Gen. Fogleman). Raptor was seen as a good generic name that would cover a range of hunting birds. Of course they didn’t realise that Steven Spielberg’s film Jurassic Park had changed the common meaning of the name Raptor to mean the dinosaur Velociraptor. Also Raptor or Raptus is Latin for ‘Thief’, ‘Siezure’ and ‘Rape’ which is probably what US appropiators think about its cost, and Lockheed Martin where Ben Rich famously suggested they would have made more money investing in CDs rather than the ATF program.