LORELAI: I always thought if he could just get it together, grow up – maybe we could do it. Maybe we could really be a family, in the stupid, traditional ‘Dan Quayle, golden retriever, grow old together, wear matching jogging suits’ kind of way.
James Danforth “Dan” Quayle (born 1947), the 44th US vice-president from 1989 to 1993 under President George H.W. Bush. He sought the Republican nomination for president in 2000, but was unsuccessful.
Dan Quayle was outspoken on “family values”, and was notorious for citing sitcom Murphy Brown, which had an intelligent, professional single mother as the protagonist, as an example of how popular culture was eroding those values, in 1993. Amy Sherman-Palladino, creator of her own show about a single mother, seems to be having a little dig at Quayle here.