The National Society of High School Scholars (NSHSS) is one perhaps less well known. NSHSS provides scholarships for high school scholars funded by various national associations and organizations. For example, the Alzheimer's Association gives two scholarship awards. One is the Scholar Award of $1,000. The other is the Leadership Award of $2,500. Another type of national scholar recognition is the Claes Nobel Academic Scholarship of $5,000. Full details about eligibility, applications and deadlines are available at the NSHSS website.

http://www.nshss.org/benefits/scholarships

The Presidential Scholars Foundation is a pretty big deal for graduating high school seniors.  It's a national honor, commissioned by the White House, for high achieving scholars in academics and community service.  It is not a scholarship program, but it is a huge recognition and achievement. 

When I was in high school in the 1970's, I was chosen to be featured in the publication, Who's Who Among American High School Students.  It was a great honor for me and not offered to everyone.  Each student was listed with their picture, honors, achievements, goals, academics, community service, extracurricular activities, and prospective career choices. It was supposed to help make us more eligible for grants and scholarships.  It's since gone bankrupt, but it was an exclusive privilege in my day. 

I would suggest a different type of recognition. Organizations like People to People or Lions Clubs choose high-school-aged students for opportunities to travel abroad, experiencing foreign cultures and meeting people from other lands. The personal growth and expanding of horizons that is made possible by such exchange programs can be life-changing for participants. Students are selected for such programs because they are outstanding citizens, open to new experiences, ready to act as positive representatives of the United States and as role models for others in their home communities.

Another one would be winning some of the more prestigious competitions that are held.  The one that comes most easily to mind is the Westinghouse competition in which students essentially do very advanced science projects.  Even participating in that is a major honor.

There are many other national honors and not all of them are academically based. For example, if you are an athelete, you can be an All-American in practically any sports and individual sports have national tournaments. If a person does well, then a high school student can gain recognition.