All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is permissible for you to give your Zakat money to your friend's mother since she has no money in order to have her operation as this is better than giving it to another eligible person who is entitled for Zakat because in addition to fulfilling the need of a poor person, you are saving a Muslim's life.
The same applies to optional charity in this case.
We ask Allaah to reward you.
Allaah Knows best.