LOUISE: So how’s that going? Are you two [Rory and Dean] still Joanie Loves Chachi?
RORY: God, I hope not.
Joanie Loves Chachi was a sit-com which ran from 1982-1983. A spin-off from Happy Days, it starred Erin Moran as Joanie Cunningham (Richie Cunningham’s sister), and Scott Baio as her boyfriend Chachi Arcola (The Fonz’s cousin), with the pair becoming singer-songwriters in Chicago. The show quickly tanked as it turned out people weren’t watching Happy Days so they could see these two lovebirds warble at each other.
Rory’s disgusted response seems to indicate that she knows the show was a disaster and doesn’t want her relationship compared to it. The show ended before Rory and Louise were born, and was never popular, but reruns were shown on Nick at Nite (apparently all teenagers on Gilmore Girls watched Nick at Nite so they could keep up with 1980s cultural references).