I am glad you posted this question, and I will also be checking out some of the suggested sites as DS is just finishing his freshman year, too.
I agree that private schools often offer more than state schools. I had three private colleges offer me great scholarships while the state school only offered me $500 per semester. I couldn't afford the state school, but had my pick of the three private schools. Go figure!
One teacher recently mentioned to me that kids can get a scholarship in an academic area even though they may not plan to major in that topic. They may have to take a certain number of classes in that subject, but not major in it. The teacher was specifically talking about music scholarships. Your child might be able to get a scholarship based on the instrument they play or their singing abilities, even though they plan to be a math major, for example. There would probably just be a requirement that they play in the college band or orchestra, sing in the choir or take some other types of music classes. She did mention a student who got an engineering scholarship and took some classes in that subject but had a totally different major. DEFINITELY worth checking into.
Less rhetoric, more cowbell!