Should I refuse the money my mother gives me, knowing that she takes money from bank interest and affirms that what she gives me is from household expenses and not from that interest?

You are not required to give up the aforementioned money, especially if your mother informed you that it is not from usurious gains, or if it is from bank interests that you both believe permissible to take, following those who have issued a fatwa (definition) to that effect. Even if the money were unlawful, its unlawfulness pertains to the one who acquired it, and it is permissible for the poor to accept it because it is one of their rightful expenditures.