Libby offers Rory a Midori sour, which she politely refuses. Rory is reading instead of boozing, no surprises there.
Midori is a bright green extremely sweet melon-flavoured liqueur made by the Japanese brewing company, Suntory, but manufactured in several countries. Made since 1964, since 1978 it has borne the name Midori, which is the Japanese word for “green”. A Midori sour is a cocktail which combines Midori, grenadine, and lemon juice.
The Annotated Gilmore Girls 