Before the slickly produced video there was one she independently produced. It has nearly 18M views on YouTube:
Wikipedia offers:
Quote:She is the daughter of domain investor and millionaire Rob Grant. Grant's management chose her stage name, "I wanted to be a band but the label I was with and the team I had around me absolutely wanted me to be a solo artist. Lana Del Rey came from a series of managers and lawyers over the last 5 years who wanted a name that they thought better fit the sound of the music."
I think it's a good stage name.
A bit on her father:
http://dnjournal.com/cover/2008/april.htm