Once upon a time, there was a punk band name generator at mp3.com, and it was hours of fun and entertainment, in the dingy computer lab on the 2nd floor of the English wing of the University of Alaska, Fairbanks campus. I poked it until hilarious things came out, and wrote down the best ones. The only other one I can remember was "Pierced Vomit", but "Azure Lunatic" came up. So I put it in the back of my brain.
Later, there was a Vampire: The Masquerade game, and I was a Malkavian, and I named her Azure Lunatic. And then she got bored of the game and wandered off with me.
I had asked my then-fiancé to help me attempt to keep the Malkavian from moving into my brainspace. That ... really didn't happen. Badly. Next thing I knew, the Malkavian had set up shop as the fourth of my constant rotation of four personality-branches, and then at about that time I needed to make a Keenspace forum account, and ... well. And then I showed up on LJ. (I have no idea when I set up that Keenspace forum account, but it was post-move, and the flirting-with-the-idea-but-not-committed was the email address, manicblue at mad.scientist.com, which is now very defunct.)
Time passed, and ... I wouldn't quite say that I grew into the name, but I molded the name around the person I became, and it and I fit each other.