Demerol

DEAN: Lorelai. I like that.
RORY: It’s my mother’s name too. She named me after herself. She was lying in the hospital thinking about how men name boys after themselves all the time, you know, so why couldn’t women? She says her feminism just kind of took over. Though personally I think a lot of Demerol also went into that decision.

Demerol is a brand name for Pethedine, a synthetic opioid medication and the most common narcotic pain relief given in US hospitals during childbirth.

Lorelai’s choosing to give her daughter her own name is another suggestion that she hoped Rory would be a “do-over” version of herself.