Is it permissible to give money earned from usurious bank interest as charity to a sister who is suffering from financial hardship, knowing that she is married and has a child?

Usurious gains are not permissible to be spent on oneself or on those one supports. One must dispose of them by expending them for the benefit of Muslims. There is no objection to giving them to a poor or needy sister with the intention of ridding oneself of haram (unlawful) money, not with the intention of charity (sadaqah), a gift (hibah), or a present (hadiyah).