Your daughters partner has to basically buy the house from himself and your daughter, at an agreeable price based on market value. They then split what's left of the equity after mortgage and conveyancing costs have been settled, give your daughter her cut ,and use his cut as a deposit when applying for a mortgage on the property in his own name. This can all be done consecutively and they need a conveyancing solicitor who will organise all the details. For these purposes you can't 'take your name' off a mortgage just as you can't add someone on, it has to be a new transaction.