Lizzy Mercier Descloux's birth name is Martine-Elisabeth Mercier Descloux.
Lizzy Crawford's birth name is Elizabeth Crawford.
Lizzy Pattinson's birth name is Elizabeth Pattinson.
Lizzy Borden's birth name is Janet Romano.
Lizzy Caplan's birth name is Elizabeth Anne Caplan.
Lizzy Calhoun's birth name is Julia Elizabeth Calhoun.
Sheila Lussier's birth name is Sheila Anne Lussier.
I think you mean Lizzy Mercier Descloux. I also tried to find her name using the same references you mentioned in your question: Graceland, female, French, punk. Let me know if this was the answer you were looking for.
Edwina Robbins's birth name is Edwina Mercier.
Merce Cunningham's birth name is Mercier Philip Cunningham.
Most people are nicknamed Lizzy from the name Elizabeth so no
Lana Del Rey's birth name is Elizabeth Woolridge Grant.
Lizzy Borden's real name is Gregory C. Harges