LORELAI: Here is the phone and some mad money. If for any reason you think you’re not going to be home by twelve, you call me.
“Mad money” is an informal expression meaning a small amount of money to be used as an emergency fund. It is especially applied to girls and women who are out on a date or a night out, and may need extra money for a taxi home in case they have an argument with or become separated from their companion/s. The expression dates to around 1920, and the origin of it has not been made clear to me; I presume “mad” in case you and your date get mad (angry) at each other.
As Rory will be driving herself (at last by now she really does have her driver’s licence), Lorelai probably intends the money to be used for car fuel, or a taxi if the car breaks down.