Surely he has deed to the land on which he built the house.

Assuming that he hasn't built it inside a city without permits, he shouldn't have a problem;
an attorney can handle the transfer for a very modest fee.

Any non-profit worth its salt will know how to do it and have the means to do so. Beware, though, they may just turn around and sell it. That may not be what your cousin has in mind, so the advice to consult with an attorney is well given. Just make sure the attorney has no ties to the non-profit entity.

There may also be tax considerations. A tax professional should be consulted as well.

Another thing, if a non profil association does use it as a home, it would help to know if it would be allowed in the area/neighborhood such as a half way house. Zoning should be listed.

Houses do not have deeds.

The deed is for the land the house sits on.

Land is generally transferred (conveyed) 'with all improvements' and that covers the structures.

Thanks for all the information. My cousin did get legal council and has decided against the donation. Instead he's going to leave it in a trust for when he passes. Thanks again.