There are a few fairly high ranked private schools that offer full-ride merit scholarships or close to it. So I think you should explore those but it can be tricky if he falls in love with a place that then ends up the most expensive. It's up to you but I wouldn't go too cheap with the best giveaways like laptops, over the top dorms, etc. If you can get the cost <$5000 or even $10000, I think it might be worth it.
A lot of this depends on your son's career aspirations. Is your son more the hard worker or more the naturally brilliant type? Is he interested in academia? Does he value doing cutting-edge research & development or would he be happy with a more traditional engineering job? Does he have either the software or finance bug? You can get the coursework at any good college so you really have to determine if research opportunities, internships, or relationships with faculty in a specialized area would be worth it. For most, I'd say it's likely not, especially undergrad. And for those it is worth it, you have to have a real passion. For instance, for those interested in academia, it's competitive and you'll likely make less money. But for certain people, the opportunity to do never before done things is worth it.