EMILY: This isn’t funny. I hardly get to see the girl and we only get to talk at dinner once a week and then it’s all about school and Jane.
LORELAI: Lane, Mom.
The fact that Emily mixes up Lane’s name and calls her Jane is a probable allusion to the MTV cartoon Daria (1997-2002), in which protagonist Daria Morgendorffer’s best friend is Jane Lane (I believe this to be source of Lane Kim’s rather unlikely first name).
Jane Lane is not Asian-American like Lane Kim, but her voice actress is Malaysian-born, and the character is drawn with some slight Asian characteristics, so that casual viewers sometimes assume she has Asian heritage. Besides both having black hair and being the less academic sidekick to an intellectual bestie, Jane Lane was a gifted artist, as a parallel to Lane Kim’s musical talent.
(By the way, if Emily only hears about Rory’s doings at Friday Night Dinner, whatever happened to that weekly phone call where she could catch up on Lorelai and Rory’s lives?)