LORELAI: So it was the uniform, huh?
MISS PATTY: Aw, it’s the Biloxi Naval Base all over again.
Biloxi is a coastal city in Mississippi. There isn’t actually a naval base there – there’s a military base for the air force though. Miss Patty might be thinking of the Naval Construction Battalion Center in nearby Gulfport, about ten miles further down the coast – the cities are so close that they share the same airport, and other facilities, and the air force and naval bases are close together.
Both these military bases provided training to new recruits in the second half of the twentieth century, beginning in World War II. It’s possible that Miss Patty, who seems to have been a New Yorker before she moved to Stars Hollow, entertained the troops at one or both of these bases, perhaps during the Vietnam War, when she would have been in her early twenties.