Your mother should consult with a lawyer to help her understand her options and to make sure that any agreements between family members are properly documented and fully understood.
indicated that your mother owned her home with her late husband. Most married couples who own a home together do so as either joint tenants or as tenants by the entirety which would mean that when your stepfather died, his interest in the home passed automatically to your mother. Did your stepfather's Will make any provisions for the payment of the mortgage out of his assets? Did his assets other than the home pass to your mother? These are important considerations.
Even if your stepbrother had an effective Durable Power of Attorney for his father while he was alive, upon your stepfather's death the Durable Power of Attorney ceased to be effective. Consequently he has no authority to act on behalf of his father or access his assets unless such authority was granted in his father's Will.
Your stepbrother certainly could agree with your mother to either give or loan the money necessary to maintain the mortgage payments, but in either case the terms of the gift or loan should be fully understood and documented. If the money were to be done as a loan, it should most likely be structured to be paid back upon the sale of the home, but in any event it is critical that a lawyer be involved to make sure that everyone understands their rights and obligations.