بسم الله الرحمن الرحيم
First you should confirm whether the mother of your Christian friend needs a second operation or not and whether she is in need of money or not? If it is confirmed that she is in need of money then you may help your Christian friend. But it shall not be lawful to give zakah money; you may support him with any other funds. Also you may give him interest money if he is poor and needy person.
Allah knows Best!
Darul Uloom Deoband
This answer was collected from the official ifta website of Darul Uloom Deoband in India.