Such a difficult decision, particularly when money is tight! I do know that kids with stressed out parents are not always better off staying in their home, even though many parents go to heroic lengths to make that possible. I also know divorced parents who moved to a more affordable home, and found that the lack of emotional baggage from the “family” home was therapeutic for everyone.
Most financial experts advise starting with a careful financial analysis. Depending on her work history and income, your friend may need a loan in her own name. If she feels strongly about staying, but does not qualify for the desired rate or amount from a standard lender, she might be able to arrange an affordable mortgage from a parent, grandparent or even the ex-spouse until she can develop a work history and save the closing costs. I wish some enterprising lender would develop a program for these situations.