The name change thing is dumb, but it makes sense.
She said she was young when she changed her name, I am assuming she was a teenager when she did it. Teenagers are silly, a teenager changing her name to Lightning and pretending to be a tough girl seems like something a "normal" troubled teenager would do. Just look at all the goth kids with their vampire names and stuff.
It is stupid, but it's something a teenager would do....
The name Lightning is sort of dumb, but look at the protagonists of other Final Fantasy games; Cloud, Squall, Tidus.....
Lightning is not so bad, it could have been worse.... like "Moon" or "Storm"
On a unrelated note: The evil black-haired girl, looks pretty cool!! I hope she joins the party .....