Is it permissible to give the money earned from bank interest to the sister supported by the questioner, or to a relative such as an uncle or an aunt?

If the bank is usurious, then the interest derived from it is forbidden usury (riba). The obligation is to repent and withdraw the money immediately, unless there is a necessity due to fear of loss, in which case it may be kept without interest. Repentance is achieved by disposing of the usurious interest by spending it on charitable causes. There is no harm in spending it on poor relatives who are not obligated to be maintained by you. As for interest taken after becoming aware of its prohibition, it is obligatory to dispose of it by spending it on charitable causes, such as giving charity to the poor and the mujahideen, along with repentance. It is not permissible to give usurious interest to those whose maintenance is obligatory upon you, such as a sister whose maintenance is obligatory.