Is it permissible for a mother who swore an oath that dates and [balah] (a type of date) would not enter her house [again to break it], and what should her daughter do, who fears that her mother will not be convinced to expiate her oath?

There is no blame upon your mother for swearing not to allow dates into her house. If she wishes to retract her oath, she must offer expiation for a broken oath: feeding ten poor persons, or clothing them, or emancipating a slave. If that is not possible, then fasting for three days, based on the saying of the Prophet, peace and blessings be upon him: "Whoever swears an oath and then sees something better than it, let him do that which is better and offer expiation for his oath." As for the hadiths that mention eating dates, such as "Eat fresh dates with dried dates" or "Eat dates on an empty stomach, for they kill worms," these are weak or fabricated hadiths. Even if it were proven that the Prophet, peace and blessings be upon him, ate a certain thing, not all of his actions, peace and blessings be upon him, are considered legislation that a Muslim is obliged to follow, especially those that fall under the category of natural customs like eating and drinking.