Well, a lot of people have already said that Axel is Lea, and Saix is Sai, but no one has said why, so I'm taking the liberty of doing so.
Axel is Lea because the definition of Lea is 'A grassland; a meadow' (go look it up!), and the first time you probably meet Axel is in a meadow on the path to Castle Oblivion (remember that mysterious cloaked figure?)
Saix is Sai because Sai is a Japanese name, and, like a lot of other Japanese names, it probably has a unique meaning. How do I know that Sai is a name? Because, in the Japanese version of Naruto (the manga; I read the scanlations), there is a character that is named Sai.
Demyx, Marluxia, and Luxord I didn't even try that hard to do what I did when I found Axel's name (scramble the letters to see if they had a definition), but I did find out that Dyme was a city in Ancient Greece. It's history wasn't that amazing, though, so that's probably not it.