Is it permissible to use the interest earned on the money the mother saved in the bank, knowing that she will borrow double this amount to cover the costs of the marriage?

The benefits obtained from usurious banks are forbidden. Your mother must repent, withdraw the capital, and dispose of the interest by spending it on the welfare of Muslims or on the poor. She can deposit her money in Islamic banks. It is not permissible for you to benefit from this money unless you are poor. However, if the bank is an Islamic one that adheres to Sharia guidelines, then there is no harm in benefiting from its profits.