LORELAI: Honey, you have power, brains, now all you need is a dimwitted, drunken or drug-addicted relative to constantly humiliate you while you serve in office.
Lorelai is probably referring to Bill Clinton’s younger half-brother, Roger Clinton Jr. He was given the codename “Headache” by the Secret Service, due to his controversial behaviour, which included accepting bribes from mobsters, and dangerous drink driving offences. He received a presidential pardon from his brother for a 1985 conviction for cocaine trafficking, meaning that although he had served a prison sentence, the crime was removed from his criminal record.
The Annotated Gilmore Girls 