Funny story - we picked the name, and thought it was fairly uncommon (which, really, it is...or was).
First off, we told someone the name and they started saying something about "The Briar Rose"; we were sitting there going "what the fuck are you talking about?" "Oh, didn't you remember that Sleeping Beauty was Princess Aurora?" < cue twin ESB-style "Nnnnnnnnnnnnooooooooooooo!!!"s >
Second, my buddy was getting married and his best pal from HS flew in (buddy is originally from the Bay Area, now living in Wisconsin); we're talking to his wife and she offhandedly mentions something about "my daughter Aurora." My wife and I sorta went "*ahem*...come again?"
I'm not sure if it's merely perception or if the name is starting to gain popularity.