You can draw up a will that will leave the house to your son. Or you can put the house in his name now, with you having a 'life estate' in the house. This allows you to live in the house until your death. You can, also, change the deed now to include both your name and his name.
Sell it to him for the amount owed against it or at a reduced basis that you offset with the gift tax. If you simply give it to him, you risk gift tax. And if you simply give the house away and then need Medicare, you risk appearing to decrease assets during the "look back" period.
In order to transfer title from yourself to your son, all you need to do is to prepare a new deed which conveys your interest in the house to your son. The deed must then be recorded at the County Clerk's office which is responsible for recording land records.
You can do this in one of two ways. You can either Quit Claim Deed it over to him immediately or you can purchase title insurance and close through a title agency. The title insurance would protect your son against liens or other issues that may come up on the future.