how can my husband get his name off a mortgage

he can refinance the loan or sell the home.

I think your husband is co-owner of the house. Most mortgage companies insist that everyone on the loan must also be on the title to the house.
There is almost no way for your husband to get released from his mortgage obligation.
He could have dad sell the house to him and your husband get a loan solely in his name. That is not what you want to do but it would at least give him control over the house. Dad would be a tenant.