This is one of those situations where it may be best to start with the end in mind. Perhaps starting to "rough in" some possible majors, potential future employment opportunities, and perhaps even locations of the country where he would like to live would make sense here as a first step. I know
@isucyfan mentioned that his son didn't know what he wanted to major in and was leaning math / science. I would advise narrowing this down further as that by itself will eliminate potential schools from the list. To me this is a much bigger decision than where to pursue employment post college. These are the decisions that put you on that path (which is huge) in the grand scheme of things.
From what I have seen, academic institution prestige is overblown, but it does matter to some (typically alumni of said "prestigious institutions"). Some of the more prestigious schools work with said alumni and arrange internships and jobs post college for students (kind of a "good ole boys club"). This can be easier to do when the student bodies are smaller, as the network is tighter. Those resources do exist to a certain extent at the bigger universities, but you are definitely more on your own.
As to academic rigor, that I think can vary as well and depend (again on where you are going, and what you are looking to do - see above). In the end, your son will be in competition with no one but himself and how he feels he is stacking up to his own expectations. I get that some academic arenas can help in stimulating this, but I would assume based on your initial post, he is a high flyer already.
Personally, I would go for the full ride at the best school for my major. Being debt free and out of college is huge, knowing what I know now. Now if what he was looking at was a niche, I would weigh that in as well (start with the end in mind). Being in the hiring game (HR), I have never made a hiring decision based on where an applicant went to school. The closest I have ever come to this is, "Did the candidate get a degree?" At that point, it is a piece of paper and not much more.