There are 2 facets to consider here. The first is who is on the deed. The deed conveys ownership, so anyone listed on the deed has an ownership interest in the property. The mortgage is the second aspect. The mortgage simply lets you know who is responsible for the money owed on the property.
You can add a name to a deed, and title or escrow company can help with this. This will add your name and interest as an owner of the property. You cannot however simply add someone to a mortgage. A mortgage is a legal document that has been recorded stating that a certain person has obtained a loan from a bank or lender, and a lien is being held against that property as collateral for the money lent. If you did not go through the qualification process at the time of obtaining the loan, there is no way to simply be added to the mortgage. To be added to the mortgage you must obtain a new loan, typically through a refinance transaction and that loan would have to be qualified in both you and your partners name to be added to the mortgage.
At the same time, keep in mind that there is no reason to be added to a mortgage, or to get your name on a mortgage unless you are looking to build your credit. If you are simply looking to have an ownership interest in the property you can just have yourself added to the title to the property, which will give you a legal right to the property.