The easiest way to recommend these volumes was for me to think about books that strongly inspired me when I myself was growing up, so here's my top ten list which I copied in my email to her. I think that these books make for excellent reading not just for 15-year-olds but for 40 and 80-year-olds for that matter. Feel free to add suggestions in the comments section.
1. One, Two, Three…Infinity by George Gamow: Physicist George Gamow’s delightful book talks about many fascinating facts in maths, astronomy and biology (Gamow's comparison of "different infinities” had blown my socks off when I first read it).
2. Microbe Hunters by Paul DeKruif: This book tells the stories of the determined and brilliant doctors and scientists who discovered disease-causing bacteria and treatments for them.
3. Men of Mathematics by E. T. Bell: This classic book does for mathematicians what Paul DeKruif’s book does for doctors. Although it romanticizes and in some cases embellishes its stories, it has inspired many famous scientists who read it and later won Nobel Prizes.
4. Almost any book by Martin Gardner is great for mathematical puzzles (for eg. “Perplexing Puzzles and Tantalizing Mathematical Teasers”.)
5. Raymond Smullyan’s “What is the Name of this Book? The Riddle of Dracula and other Logical Puzzles” is another absolutely rib-tickling book on puzzles and brain teasers. What is remarkable about Smullyan's volumes is that many of his apparently silly puzzles are not only quite hard, but they hint at some of the deepest mysteries of math and logic, such as Gödel's Theorems.
6. "My Family and Other Animals" by Gerald Durrell: This delightful book talks about the author’s experiences with animals of all kinds while vacationing on a small Greek island with his family.
7. I would also recommend science fiction books by H. G. Wells if he likes fiction, especially “The Time Machine” and “The War of the Worlds."
8. "Surely you’re joking Mr. Feynman" by Richard Feynman: Feynman was one of the most brilliant physicists of the 20th century, and this very funny autobiography documents his adventures in science and life. Even if he doesn’t understand all the chapters it will give him an appreciation for physics and how physics can be fun.
9. "Uncle Tungsten: Memories of a Chemical Boyhood" by Oliver Sacks. Oliver Sacks was a famous neurologist but this book talks about his exciting adventures with chemistry while growing up.
10. "A Brief History of Time" by Stephen Hawking. Some of the chapters may be advanced for him right now but it will give him a flavor of the most fascinating concepts of space and time, including black holes and the Big Bang.
11. "King Solomon's Ring" by Konrad Lorenz. This utterly entrancing and hilarious account by Nobel laureate Konrad Lorenz talks about his pioneering imprinting and other experiments with fascinating animals like sticklebacks and shrews. The story of Lorenz quacking around on his knees while baby ducks follow him is now a classic in the annals of animal behavior.