Full disclosure: I'm currently a high school student, which may be a positive or negative thing.
My first suggestion would be a course on set theory. Starting with naive set theory, you examine the diagonal argument, paradoxes, and early developments. Then you use that to motivate axiomatic set theory (perhaps ZF), derive Peano Postulates, prove Cantor-Schroeder-Bernstein, survey cardinal arithmetic.
If they've seen computational side of calculus, another idea could be to do introductory analysis course. Assuming little but rational numbers, you could construct real numbers, show their uncountability; then do the calculus they were taught, proving everything on your way.
Other suggestion with which, I feel, one cannot go wrong, is elementary number theory. I would stress the prime number theory, proving Bertrand's Postulate and stating prime number theorem.
Full disclosure: I'm currently a high school student.