Answer B is better.
The two sentences are both grammatically correct, but the meaning of A is unclear. Whose birthday is it — the mother’s or the daughter’s?
This problem also comes up in spoken language, but when we’re speaking, we have other clues to help understanding, such as intonation and gesture.
Our tip: Take extra care when writing a story that involves two men or two women, as it might not be clear who the personal pronouns “he” or “she” refer to.