Since listing my favorite math books of 2015, I was recently reminded that the new Barry Mazur/William Stein volume on the Riemann Hypothesis is due out at the end of January 2016:
http://amzn.to/1lZuxxb
Too late for Christmas, but what a great start to the new year. David Mumford calls it "a soaring ride." I suspect once out, this short volume will be THE book (out of many available) to introduce folks to possibly the most important unresolved, far-reaching conjecture in all of mathematics. (...Perhaps I already know my favorite book of 2016!)
Meanwhile, I just obtained a couple of fine prior books on paradoxes, and feel safe recommending both well-ahead of finishing them. Roy Cook's 2013 "Paradoxes" is a good, fairly standard treatment of what I believe is one of the most important topics in all of math/philosophy, for bright high-school-level-and-above students.
Stanley Farlow's 2014 "Paradoxes In Mathematics" looks to be an especially wonderful introduction to several of the classics for middle-to-high-school students particularly, in breezy but broad-covering fashion. I was previously unaware of this succinct little volume from Dover, and am delighted to have stumbled upon it. Again a great stocking-stuffer for that distinctively math-inclined youngun on your list.
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