A few projects that high school students can work on:
MATH:
Some projects that I worked on when I was in high school (circa 2010):
- Pythagorean triples: Finding out general formulas for (primitive) Pythagorean triples, possibly by inspecting lots of examples. Generalizing to general sums of squares. Reference: Burton - Elementary Number Theory
- Repeating decimals: Divide the integers 1 to 6 by 7. Observe the repeating decimals. Does this happen for division by any number? Reference: Projects 16 and 23 here
I worked on the following with 6 classmates for the Biennial Exhibition in our school in 2011:
- Zeno's paradox: One of Zeno's paradoxes.
- Puzzle of 15: Remember this Windows Vista/7 gadget?
- Monotone sequences: Can you create a puzzle based on this beautiful theorem?
- Geodesic dome
For more, look at this beautiful collection compiled by Joseph Malkevitch.
COMPUTER SCIENCE:
If you are into programming you can try participating in programming contests on CodeChef, HackerEarth, etc.
You can try writing codes for the following games:
- Sudoku (try writing a solver)
- Tic-Tac-Toe
- Connect Four