My mother has no POA. My step father wants to get a reverse mortgage, but she can not sign her name and doesn't know her name. What should they do?

Guardianship and/or conservatorship would be best. You probably need doctor's statements, just very brief that she is incapacitated, and she may have to be given the paper that states a process is under way and have a chance to contest it, which she will not likely be able to do. You might be able to find an elder care consultant who can make sure the reverse mortgage is the best idea, and what kind of trusts and financial handling need to be done (and not done) to assure she can get Medicaid benefits, etc. with or without a spend-down, They have programs where they pay part of the costs of long term care to help the well spouse stay in their own home. It varies from state to state and there are a lot or rules like "look-back" and "gifiting" and different types of trusts, which muct be done correctly.

Does your mother have diagnosed dementia? If your mother has been deemed mentally incompetent then she cannot sign any legal document. I am not even sure if you can get her PoA now. The only way may be a guardianship. I would highly recommend contacting legal aid in your area (it's free) to get some advice. Do not let her sign anything until you do.
Reverse mortgages work for a lot of people, but they come with a lot of restrictions too...again, seek advice.
Is your Mom eligible for Medicaid?