I would recommend highly "What is Mathematics" by Richard Courant and Herbert Robbins. This is very accessible book for high schoolers who are keen and interest

I would recommend highly "What is Mathematics" by Richard Courant and Herbert Robbins. This is very accessible book for high schoolers who are keen and interested in mathematics, and will expose the reader to a broad array of topics and pique the interest and awaken the imagination and instill the beauty of mathematics. This in turn can drive the reader to find out more and fall in love with the subject. I would second this by "Concrete Mathematics" by Graham, Knuth and Patashnik. This is actual university course book with very formal proofs and theory, but the subject matter is still largely accessible to serious high school students and demonstrates beautiful reasoning examples throughout. It is also very practical book, after covering techniques in this book, one can often times calculate exact sums of infinite series quicker than estimating their bounds. If your high school student decides to study math at university level, the techniques and skills taught in this book will prove invaluable in broad areas of study.