Is it permissible to make the intention of an ongoing charity (Sadaqah Jariyah), such as digging a well, for the deceased alone, or for her and for the living together?

A donation (ṣadaqah) is not binding merely by intention. Therefore, you are not obliged to dig a charity well on behalf of your mother. It is permissible for you to change your intention so that the donation is on behalf of your deceased mother and your living father, as the reward for charity benefits both the living and the dead.