What should a wife do regarding her husband's mother's request to light candles for her daughter on the occasion of the Prophet's birthday (Mawlid), knowing that this celebration is an innovation (bid'ah), and how should she deal with her given the mother's insistence and old age?

Lighting candles on the occasion of Mawlid is an innovation (bid'ah). There is no need to argue with the old woman about Mawlid being an innovation. One can be kind and gentle with her, and visit her on days other than Mawlid. If she asks about lighting candles, one can use subtle phrasing (tawriyah) and allusive remarks (ma'areed).