RORY: He’s my … gentleman caller.
LANE: Okay, Blanche.
Rory is referring to Tennessee William’s 1944 play The Glass Menagerie, where the mother Amanda, a faded Southern belle, is obsessed with finding a suitor, or “gentleman caller”, for her daughter Laura, who has crippling shyness.
I believe Lane’s response is likely a reference to the sitcom The Golden Girls, which originally aired from 1985 to 1992. It featured four older women sharing a house together in Miami, Florida; Blanche Devereaux (played by Rue McClanahan) was a slightly man-crazy Southern belle who’d grown up on a plantation. The character was inspired by Blanche DuBois in A Streetcar Named Desire, and Scarlett O’Hara in Gone With the Wind.
The Golden Girls was a ratings winner which received numerous awards, and is considered one of the best TV sitcoms of all time. It continues to gain new viewers in syndication, and has aged extremely well.
EDIT: Edited in response to reader Holly, who suggested it was more likely that Lane was referring to Blanche in The Golden Girls than to Blanche in A Streetcar Named Desire.