As far as I know, Lightning is her codename. But it's not her real name. We "have to look forward" to the story surrounding her real name and stuff. I'm smelling another main-character-with-troubled-past-thing.

If you wondered where I got this, it's from an interview with Toriyama. Lemme see... Ah, here it is, just scroll down a bit to the part where they ask about 'Lightning':
http://www.ff-xiii.net/boards/showthread.php?t=2817