My daughter and fiancée are buying a house, which would be the easiest and least expensive, adding my daughter to the mortgage after they are married or changing her name after they are married.
My daugther's fiancée grandmother is co-signing for them. My daughter will be paying half of the monthly mortgage payments.

I don't understand your question. If your daughter is not on the original mortgage she can't simply "add" her name. A new application would have to be filed and approved.

As far as changing her name - I assume you mean on the mortgage - that is not necessary. She is the same person, same SS number, no matter what legal name she is using.

So the question is whether she and her husband should get a mortgage in both names - or just his name?