What a fantastic wedding present!
I think you and your father should definitely pursue refinancing. Ask your mortgage professional about the possibility of transferring the home into your name via gift of equity, if that's what you both want.
Otherwise, your father refinancing it as an investment property is a possibility.
The lowest rates are for owner occupied properties, but rates are down on all loan programs.
I recommend that you get a Good Faith Estimate from a mortgage broker and a bank in order to compare loan offers.
Best wishes to you,
Carolyn Warren