Some private schools do offer scholarships for students. I would contact the private high schools in your area and speak to their enrollment and financial aid staff to see what they have available for potential students.

If you reside in the United States, SchoolFinder can help you locate the schools' contact information: http://www.education.com/schoolfinder/  
First search by city, state and/or zip code, then select the "High Schools" tab in the navigation under the map.

Here are some additional resources you may find helpful:

Associated Content article, "Scholarships for Adopted Children"
http://www.associatedcontent.com/article/112867/scholarships_for_adopted_children.html

About.com listing of Private School Scholarships:
http://privateschool.about.com/od/scholarships/Scholarships.htm