RORY: He’s my … gentleman caller.
LANE: Okay, Blanche.
Rory is referring to Tenneessee 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.
Lane confuses it with another play by the same author, A Streetcar Named Desire (1947), whose lead character is another faded Southern belle named Blanche DuBois. Lane may know the play from the 1951 film version.
A subtle reminder that Lane is not on Rory’s literary wavelength. Notably, Rory does not pick her up on her error – she doesn’t try to change or bore Lane the way she did with Dean.
The Annotated Gilmore Girls 
I believe her reference was actually the character Blanche from The Golden Girls.
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