It depends on a number of factors, many of which are outside of your control. Given the current environment around education, schools are tightening up the merit base stuff quite a bit, and for some schools, attracting white males is not a high priority.
Not knowing what your financial situation is, you can look at two other scholarship options: need based scholarships and aid and private scholarships. Need base scholarships are based on your income as a parent, and depending on the situation, you may qualify. Schools like Notre Dame and Vanderbilt commit to providing 100% of need based aid. Contact the schools admissions office to learn more about that.
Private scholarships are provided through a number of places so you should look into those. His high schools counseling center can probably assist you with that. Also niche.com has some information about free scholarships.
Finally, if the school wants him, and it’s a matter of money, you can always go back to the school and tell them you can’t afford it I’ve had a number of friends and acquaintances do this and schools will usually move on the price. Also depending on the school, it might be to your benefit to wait to accept the offer because usually the longer you go the more incentive so throw on the table.