Symbolism, Imagery, Allegory
Despite the fact that it's an important symbol in the novel, the cup of stars never actually appears in it. Not once.
Eleanor first hears the cup of stars mentioned by a mother at a country restaurant. The mother's little girl refuses to drink her milk from any cup that isn't her cup of stars. The mother tries to convince her daughter to drink her milk from another glass, and Eleanor thinks:
Don't do it, […]; insist on your cup of stars; once they have trapped you into being like everyone else you will never see your cup of stars again; don't do it. (1.61)
The cup of stars is the girl's test, and it represents for Eleanor her ability to be her own person, to want the things she wants, and, perhaps most importantly, to do these things against her mother's wishes.
As the story progresses, the cup of stars appears several times in Eleanor's fantasy home—you know, the one she tells Theodora she has but has really just made up. The cup's appearance in her make-believe home suggests Eleanor's desire to obtain what the little girl had: individuality, independence, and freedom from what others desire her to be.
Of course, the irony of Eleanor's desire is that it eventually turns on her. When Eleanor asks Theodora if they can move in together, Theodora rejects the offer, not caring about her friend's desires or needs. Theodora, it seems, has her own cup of stars, and she hasn't offered Eleanor any.