Is reading Surat Ar-Rahman and Surat An-Nur over water, then bathing with it daily with the intention of hastening marriage, a correct act or an innovation? And how can one convince one's mother of that?

There is no evidence that specifies Surah (definition) Ar-Rahman and Surah An-Nur for facilitating marriage, and believing this without religious proof is an innovation (bid'ah). There is no basis for the mentioned method of recitation. However, supplicating for marriage after reciting the Qur’an is a good deed. Convince your mother that marriage is in the hands of Allah, and that obeying Him, supplicating frequently, and earnestly persisting in supplication are among the most beneficial means to attain one's desired outcome.