The main thing I (speaking of course only for myself) would want students to learn from this exercise is that it is possible to study mathematics, as opposed to using mathematics to study other things. So I'd steer clear of applications to voting or music or gambling and steer them toward writing about prime numbers (say, the infinitude of primes, or the fundamental theorm of arithmetic, with applications to the irrationality of the square root of 2) or topology (konigsberg bridges, or the euler characteristic of a sphere, or the connect-three-houses-to-three-utilities problem, except without trying to make it sound like applied math) or geometry (platonic solids) or set theory (uncountability of the real numbers) or quadratic forms (e.g. sums of two squares).

These are of course all terribly unoriginal ideas but what matters is that they're new to your students, not to us.

A student who could explain what a group is, and give a few examples, and prove that the identity element of a group is unique, would have accomplished something quite substantial for this age.