Highly selective, expensive private schools tend to emphasize need-based aid.
However, some lesser-known or slightly less selective private schools do offer merit scholarships in the $15K-$25K range. Examples: Oberlin, Hendrix, St. Lawrence, Lawrence U., Agnes Scott, Rhodes … Case Western, Tulane, URochester, GWU, Pepperdine, Villanova.
Highly selective private universities that have offered merit scholarships (in the recent past, at least) include Vanderbilt, Rice, Duke, UChicago, UND. These awards may be few in number or small in average amount.