LORELAI: Here. [hands Rory a drink]
RORY: What is it?
LORELAI: Shirley Temple.
RORY: What are you drinking?
LORELAI: A Shirley Temple Black.
(Lorelai lets Rory smell her drink.)
LORELAI: I got your Good Ship Lollipop right here, mister.
Shirley Temple (1928-2014) was a multi award-winning box office-topping American Hollywood child star who began her career at the age of three. In 1934, she became famous starring in the film Bright Eyes, written specifically as a vehicle for Temple. On the Good Ship Lollipop is Temple’s musical number from the film, which became her signature song.
Later in life after leaving Hollywood, Shirley married a wealthy businessman named Charles Black so that her name became Shirley Temple Black. Strongly interested in conservative politics, she was US ambassador to Ghana and Czechoslavakia, and was the first female Chief of Protocol of the United States.
The Shirley Temple Lorelai hands Rory is a non-alcoholic cocktail, made for children and named after the child actress. Traditionally it’s made from ginger ale with a splash of grenadine, decorated with a cocktail cherry. Modern versions may be made with lemonade, lemon-lime soda or orange juice. Shirley Temple herself disliked the cocktail, finding it too sweet.
Lorelai indicates that her own drink is the grown-up version of a Shirley Temple (or just a very grown-up cocktail). It is possibly a Dirty Shirley, a Shirley Temple with vodka or rum added to it.