The Joint Strike Fighter, aka the F-35 Lightning II, was named after the P-38 Lightning, a long-range escort fighter that saw extensive service over the Pacific during World War II. Lockheed Martin (or just Lockheed, back then), is responsible for both aircraft, and they set up a little photoshoot where new school met old school.
The Air Force decided to name the F-35 Joint Strike Fighter "Lightning II" in honor of the heavy strike fighter that had the most air-to-air kills in the Pacific during WWII, the P-38 Lightning. They also decided to name it "Lightning" because, according to the good old USAF:
"The Lightning II name also draws parallels with a formidable force of nature. Like lightning, the F-35 Lightning II will strike with destructive force. The stealth characteristics of the jet will allow the F-35 to strike the enemy with accuracy and unpredictability; when the enemy finally hears the thunder, the F-35 is long gone."
There's actually another Lightning that was thrown into the naming mix as well. The English Electric Lightning, a Cold War-era supersonic jet fighter, was built by a predecessor of BAE systems, a British contractor also involved in F-35 production. The EEL, as it was called by nobody but me just now, was the first fighter jet capable of supercruise, which is the ability to fly faster than sound without having to kick in fuel-gulping afterburners. It was quite a looker, too, with engines mounted vertically in the back:
You can see a bigger pic of the English Electric Lightning over at Airliners.net, and we've got a couple more shots of the two Lockheed Lightnings for you, just below.