Question regarding Hebrew names - is there a general rule for when we use Ben/Bas Mother Vs. Ben/Bas father? For example I know for an alliyah to the Torah reading, it’s my Hebrew name Ben my Father’s Hebrew name and when someone asked to Daven for friends or family it’s the Hebrew name Ben/Bas the mother’s Hebrew name. I was wondering about other instances when Hebrew names are used and if there is a general rule?

Although there are varying customs, the general rule is, as in most cultures to this day, the family name goes by the father, "by families following their fathers' houses" (e.g. Bamidbar 1), and that's what's used throughout the Tanach [except for rare exceptions, e.g. Yoav, Avishai and Asa'el ben Tzruya, probably because she was King David's sister (Radak, Shmuel ii 3, 39), or others explain that her husband had passed away young, or that she was a particularly great woman, Iggeret R. Sherira Gaon). On the other hand, when we wish to evoke mercy, the way of the world is to turn to the more feminine aspect (e.g. Tehilim 116, 16, "Please, O Lord, for I am Your servant; I am Your servant the son of Your maidservant..."), so when one is God forbid sick, we pray using the mother's name.