I sincerely doubt the housing authority will take him seriously. It's not intended for able bodied young men who can get a job. Furthermore, there are usually wait lists for housing, and they are not going to bump off indigent disabled people and mothers with children so a teenager can have a publicly funded apartment.
He needs to advocate proactively for himself. He's not a child. There are thousands of college students facing this particular dilemma every semester. He needs to do what they all do...advertise for room mates, get involved on the campus listserve looking for people in the same situation, and then get an off-campus apartment with other students with whom they can split the cost. I am sure his college has a housing department that works with students. He needs to call them.
Going to college is about being a grown up. If he got into school, he's obviously smart enough to figure this one out!