For tarawih at home, my husband wants eight rak'ahs but my mother-in-law wants twenty. How do we resolve it?

Both practices are followed among scholars and Muslims, so it is not something to argue about within the family. Tarawih is a voluntary prayer, and what matters more is focus and harmony at home. One simple solution: pray eight rak'ahs together in congregation, then whoever wants to continue to twenty can carry on, either in congregation or alone. The husband can also honour his mother-in-law's custom by joining the twenty rak'ahs on the nights she is with the family. Avoid mocking other people's practice in front of the children. Teach them that differences in voluntary worship can be embraced with good manners. See differences in fiqh practice.