Lala's full stage name is Lala Lamazing Grace:
"Lala Lamazing Grace is a singer with a fake French accent and a fake French past who is forced to confront the mother and the daughter she left behind in order to look for fame and fortune."
However, during the play we find that Lala's real name is Sadie:
"It’s not clear where the sketch about LaLa, a glamorous singer who used to be named Sadie, is going until someone tells her there’s a child in her closet. When the child is revealed, she’s a beautiful, dark-skinned little girl in pigtails, the person LaLa used to be." New York Times theater review
In the play, Lala says:
"First off, no one calls me Sadie. Sadie died the day Lala was born."