A friend of mine is recently divorced and her ex signed over the house to her through a quitclaim deed. Is refinancing her only

Yes. She must refinance the mortgage.

Unless the final judgment requires your friend to refinance the house, she isn't required to do so. If it does require her to do so, she may be able to assume the debt without going through the refi and losing her interest rate. She should talk to her bank to see what other options are available.

The safest way. You could pay off his loan with him getting all the interest deductions and making it up to you some other way maybe.