Research done by high-school students

In 1988 in the IMO, Australia decided to use the following question:

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Prove $\frac{a^2+b^2}{ab+1}$ is a perfect square.

The problem was proposed by Stephan Beck, West Germany. No one in the committee was able to solve it. Two of it's members were George Szekeres (Erdős number of 1) and his wife, both famous problem solvers and problem creators. The problem was then sent to 4 prominent number theory researchers and they were asked to work on it for six hours. None of them could solve it in this time. The problem committee submitted it to the jury of 19th IMO marked with a double asterisk, which meant a super-hard problem, possibly too hard to pose. After a long discussion, the jury finally had the courage to choose it as the last problem of the competition.

Eleven students gave perfect solutions. The solution to the question used a new technique in problem solving that had never been used before. However 11 high school students were able to surpass prominent number theorists in their own field by solving the question. The technique used for solving the problem is called Vieta Jumping.

I'm not sure this is really what you're looking for, but Britney Gallivan, then 16, disproved the famous claim that it was impossible to fold a piece of paper in half ten times, by folding one twelve times. She also came up with a model that correctly explained the limit, and predicted how big the original paper would have to be to be folded $n$ times.

Archive of page about Gallivan from the Pomona Historical Society

He may not have been in "high school" but he was certainly at that age when he "was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem." I'm of course talking about Galois, whom I am amazed has not been mentioned yet.