Fractional calculus is possible for an advanced high school student to "discover", but probably not to "develop" the theory very much. Doing so, at the very least, requires some complex analytic properties of the Gamma function which we use as the meromorphic continuation of factorials onto the complex plane.
If you already know basic calculus, perhaps your project could be to write an essay that places some of those calculus concepts on a firmer, more rigorous groundwork (ie, introduce yourself to basic real analysis). Apostol's Calculus Vol.1 does this quite well.
Is it possible to interpret fractional calculus in a physical sense like the first derivative is the rate of change, but if it's fractional, what does that represent?