I was thinking along the lines of Abstract Algebra too, but I considered that it might just be too upper-division for the average high-school student.
But I do feel that any high school student motivated enough can at least tackle some of the basics. The right book is important too. The 900-page textbook I am using for my graduate level course is probably not the best idea.
A text book that isn't too dense should be fine. I think problems might arise with constructing new groups, though, like modding out by the kernel of some homomorphism, Field of Fractions, etc... Students will need a grasp on Set Theory for that.
Last March, I introduced Dihedral groups to my brother's 3 older kids, 11 to 13 at the time, and they were perfectly capable of filling out a Cayley table for the groups of symmetries of the equilateral triangle and square and had some fun with it. It wasn't difficult for them to notice that each element appeared in any given row or column exactly once, aside from the headings.
But perhaps the book I might recommend is my textbook for Sets and Logic:
Mathematical Reasoning: Writing and Proof by Ted Sundstrom
It would be great for teaching students that writing and mathematics are not necessarily two different animals.