Looks like it was officially announced about a week ago the F-35 is the Lightening II.

http://www.af.mil/news/story.asp?id=123022931

Given the circumstances though, it couldnÆt be an easy plane to pick a name for. USAF planes are typically birds, while US Navy plane have typically been cats, and this plane will be used my USAF, USMC, and US Navy, as well as several other countries. I would have been impossible to please everyone.