Saul was a bit complicated, but Elijah isn’t. The information given on the data base seems to be correct, except the name doesn’t necessarily have to mean “MY God is Yahwe”, it can also be translated “Yahwe is (the one and only) God”. But “My God is Yahwe” would still be the better translation, because the name is a more or less personal confession to the God of Israel. May others believe in other gods – mine is Yahwe. Elijah himself asks this very question to the people of Israel: “Make up your mind! If the Lord is God, worship him; but if Baal is God, worship him!” (1. Kings 18,21).
Unlike with many other names, the bible does not tell anything about the etymology or the name giving motive of Elijah. He is just introduced as “Elijah from Tishbe in Gilead” (1. Kings 17,1). We can assume that Elijah was his real name, because if it hadn’t been, the authors of the bible would probably have made some remark on that. But his name couldn’t have been picked better, because “Eliyahu” (“-hu” means “he”) turned out to be something like the “programme” of his life.
You can find out more about Elijah's life and about his meaning in Judaism and Christianity, if you check the net (there are tons of sites on bible stuff). But you can start by reading 1. Kings 17 to 2. Kings 2.