Fit is really only something you can tell by being on campus. Your question is more about narrowing down the field to get to the point where you are looking at fit. In this internet age it is both harder and easier. More difficult because you can learn of the existence of schools you might otherwise never have had to consider. Easier because you can get the answers to very, very specific questions without ever setting foot on campus.
At this stage, I would ask make a list of Must Have’s and Can’t Haves, and go through the schools that way. If he cannot stand the idea of all-boys, cross those off the list. If a school doesn’t offer the right math classes, out it goes. If he will only go to a school with block scheduling, figure out which ones have it and keep them, discard all the rest. If you need a direct flight or a particular driving distance, purge anything that doesn’t meet that criteria and keep the rest. Keep doing that until you get to a manageable number of schools to go deeper with tours, formal interviews, etc.
You might want to look at Western Reserve Academy in Hudson OH. Has math up to Multivariable Calculus and Linear Algebra. Smaller school. Healthy endowment size. Variety of class formats and teaching styles. Good soccer coach. Interesting use of Saturday schedule.