How much will it cost to add my name to my partners existing mortgage?

It shouldn't cost ANYTHING. Just ask the lender. But I think that's not quite what you want. Because Adding you to the mortgage only obligates you to pay the loan. You still won't have any ownership interest in the property.

What I think you want is to be added to the DEED. And for that you need to file a quit claim deed from the current ownership to the new ownership. However, the mortgage holder may not allow that. So you should start with them for permission to change the deed.

The costs will vary but should be minimal.