RORY: Fine, forget it. Should I put your name on Grandma’s present?
LORELAI: Yes, sign it the inn keeper formerly known as her daughter.
Lorelai is making a reference to the American pop singer Prince (1958-2016). In 1993, after a battle with Warner Bros., who refused to release his huge backlog of music at a steady pace, Prince changed his name to a symbol which combined the symbol for male with that for female, and which he later copyrighted as Love Symbol #2 – it had already featured on his 1992 album The Love Symbol Album (it has to be called that as the actual name cannot be said).
Because Prince’s new name was unable to be spoken, he was referred to as “The Artist Formerly Known as Prince” until May 2000, when he reverted to his real name of Prince (he was born Prince Nelson).
Lorelai is saying that her relationship with her mother has come to a complete and bitter end, much like Prince’s relationship with his record label – although in fact Prince was to re-sign with Warner in 2014.
Lorelai refers to herself as an innkeeper rather than an executive manager because she is still thinking of the Christmas pageant, where the innkeeper refuses entrance to Mary and Joseph, forcing them to spend the night in the stable, where Jesus is born. She likewise doesn’t feel like admitting her family into her life.