Is it permissible for the mother to give a beggar an amount from the bank interest and donate the rest, or is that not permissible for her? In this case, is it permissible for the beggar to borrow from her and then repay her the amount in installments and donate it to the poor?

It is not permissible to deposit money in a usurious bank except out of necessity to preserve the money, even if the depositor disposes of its interest, because doing so constitutes assisting a usurious institution or entering into a usurious contract with it. Usurious interest must be disposed of by spending it on charitable causes and public interests or by giving it to the poor and needy. Its owner will not be rewarded for it with the reward of charity because it is not his property; rather, he will be rewarded for obeying Allah's command. If the questioner is poor, it is permissible to give him from this money an amount sufficient for his needs, like other deserving individuals, and he is not obligated to return it, although it is better for him to give a similar amount in charity if Allah expands his provisions.