How to become my mom's Power of Attorney?

Do some research. There are different types of "POA" and many loopholes and other catches. If there is money available think about having it written into the legal documents that you will be paid or reimbursed. Also, if you are unable or something happens who would step in next.
This website has some excellent advice under this topic. I wish we had read it before doing our paperwork.
As with ALL legal issues see a lawyer. Elder law is different from state to state.

I suggest you start with seeing an elder law attorney. You may want to have a will, living trust all done at the same time.