Lorelai has this song playing in the background while she waits for her pizza to arrive, until Dean knocks on the window.
Christmas Wrapping is a Christmas song by American new wave band The Waitresses. It was written for a 1981 compilation album called A Christmas Record, put out by ZE Records, a New York “hipster” record label. The album was re-released in a “special edition” in 1982. Christmas Wrapping didn’t chart in 1981, but reached #45 in the UK after the 1982 re-release. It has often been included on Christmas albums in the UK, and re-recorded several times.
The song is about a busy single woman who decides not to bother with Christmas that particular year, turning down invitations and planning to spend the day alone. On Christmas Eve she runs into a man she had earlier made a connection with, and winds up having a happy Christmas with him after all.
Lorelai is also planning to give the Christmas party a miss to be alone for the evening, but she does end up spending time, not just with a man that she likes, but with her daughter and parents as well. The song is letting us know that a happy ending is ahead.
(Lorelai’s choice of Christmas music is also a nice contrast with her parents’)