I don't suppose anyone can make meaningful specific suggestions as to what the student might read, since it's probably best for you to suggest a book in a topic you love and are excited about, so that you can pass on your enthusiasm and insight most effectively.
If you think the student could begin a meaningful research problem with you in the timeframe you have to work with him (and if you think you have the time to give), I would suggest seeking a problem which (a) is worth your time research- and career-wise to think about (and thus publishable if finished), (b) has an experimental/exploration aspect, perhaps involving difficult but doable computations either by hand or on computer, and (c) has a wide spectrum for success, ranging from producing new conjectures based on computational evidence to theorems relating to existing literature.
I would then choose a focused literature curriculum for the student based on preparation for the particular problem, rather than geared towards breadth/culture. No matter how focused you try to be, you will find he has a nearly insurmountable amount to learn, in order to even get started. That will be a lot of work for you both, but very useful to the student. I would plan to work through problems and examples in person with the student when you are first getting started, and taper off in some systematic way, so that they have a chance to make progress on their own, but after having seen you be confused and having been confused with you, so they know how that works, and that it is okay.
If you are successful, and the student gets even a glimpse at real research and discovery, it will sustain him through a lot of dull lectures in his future education. And he will have some insight into the process of turning intuition and computations into definitions and theorems that most of us get only in grad school. Of course, item (a) is already not so easy for oneself or a graduate student, and (a)-(c) will be very challenging indeed. But it's a guideline. I think (a) is important for your morale and his, while (b) is a logistical matter - gives you a place to start. (c) makes sure you have a place to end.
Note that if a high school student does any research, this can be submitted to various competitions like Siemens, Intel, ISEF, state and regional science fairs, etc. The experience of preparing for these is very worthwhile, in my experience. If nothing else, it provides deadlines, which force the student to write things down and explain himself.
In addition to the excellent suggestions of Qiaochu, I'd also note that some REU's under special circumstances will take high school students (basically only if they are prepared far better than the average undergrad applicant, as I understand it).