Lana Del Rey, aka Elizabeth Grant, used to go by the name of Lizzy Grant when she first started out. You can still find some of her earlier gigs online, like this one from June 2009:
But Lizzy Grant had one album that really didn't sell very well, and it was then that she changed her name to Lana Del Rey. But how did she come to this esoteric name? She told MTV's Buzzworthy that "I knew that I wanted a name that sounded sort of exotic and reminded me of the seaside on the Floridian coast. ‘Lana Del Rey’ sounded beautiful." She was spending a lot of time in Miami talking to her friends in Spanish, and it was through those conversations that she came up with her new stage name.
Watch a short video here of how Lana Del Rey came to choose her name:
Hope this helps!