I would argue that a math text is usually not an appropriate graduation present, even for somebody very interested in mathematics.
Just because he likes math, it doesn't mean that he likes reading books on it in his free time
I am an avid book reader myself, and have shifted heavily to non-fiction in later years. I have discussed books with friends frequently, and find out that they are much less inclined to read than I expect. And when they do read, they do it for entertainment, and if they start a non-fiction book, they seldom have the motivation to finish it. These people are mostly STEM graduates, quite a few of them have a Ph.D. too. Even the ones who do read non-fiction now (in their 30s and 40s) say that they developed the taste late.
As you can see here, many people do like reading this kind of book. But as far as I can tell, many more don't like it. And don't forget that math.stackexchange is not a place representative of all people who like maths, it is a place representative of people who like spending their free time on a web site about maths, which is probably a subset of the first ones. And I argue that it is a smallish subset.
Even if he likes math and likes reading books on it, it doesn't mean that he wants to get one from you.
First, book choice can be a rather personal matter. It is very hard to meet someone's taste, even if you know the general direction of what they like.
But there is this small problem with learning: All people love to learn, nobody likes being taught. (I don't remember whose aforism is this, maybe Churchill?). If you see curiosity and learning interest in a younger person, it is good to encourage them, help them discover the information they already thirst for, etc. But by just giving them a book because you think the world would be a better place if everybody has read it, you come across as condescending, the kind of 19th century educator who thinks it is his high moral duty to bring children on the path of rightness.
From your question, it seems that you fall at neither of these extremes, and are probably closer to the first case. But it still isn't close enough. You don't know him, don't know exactly what he is interested in. Maybe he isn't taking calculus because he loves calculus, but because he feels a high pressure to get into a good college. And he doesn't know you, and doesn't know your motives; it is easy for him to assume that you are the condescending kind even if you aren't. Even if he doesn't, choosing a book he might be interested in has a high chance to fail.
Even if he likes math and would read books on it, this is not necessarily the right occasion.
He just made it through school and wants to feel that he achieved something after making it through a pile of textbooks. He needs a feeling of closure, even as an intermediate step. Yes, there is college somewhere in his future, but he probably wants some time to stop, look back at what he did, and smile, before getting back on the treadmill. Having finished one level of something only to be presented with the next, higher level immediately is a sure way to demotivate somebody from ever achieving something. No, I have no guarantee that he feels this way, but you have no guarantee that he doesn't feel this way. And I have seen many people, including bright ones open for education, who are this way.
Besides, graduation is a social occasion. No matter how much of a nerd he might be, even if he is internally proud of being a nerd, he probably won't want his classmates to remember him as "the nerd who got textbooks for graduation day". Teenagers are very sensitive about their image in their class.
When you start with the fact that math belongs to his favorite school subjects and conclude that he will like receiving a math text as a graduation present, you are making a lot of assumptions, and there is a high risk that at least one isn't true. He might be one of the people who will truly enjoy a math book for graduation day, but the chance that he won't is quite high.
For your decision, you can try getting better intelligence. Let your girlfriend find out through the family grapewine how likely he is to appreciate this kind of gift. If you hear that he has earlier purchased math books for himself (which were not required for a course), then this is a very good indicator that your gift will be well received. Mentioning that you knew that he reads math books for fun will also clearly remove you from the "condescending old relative" zone.
If he hasn't done this, or you can't find conclusive information, consider some other, more accessible, gift. When you meet him, you can discretely evaluate his reading interest yourself, and if the result is positive, you can still get these books for him for Christmas.
If you don't go for a math book now, here are some other options which are math-themed:
- a popular book, as you mentioned. There are a few good suggestions in this thread already. Beside things like Goedel Escher Bach, you can also get a puzzles book like one of Smullyan's. (Yes, it is still a book. But it is entertainment, and read in a very different context).
- a puzzle, or set of puzzles. Most mathematicians I know love physical puzzles. Not the jigsaw ones, but the kind where you have to find out how to take a box apart, etc. Or just a beautiful chess set.
- an entry for some event. This works a bit better with people interested in engineering, as there are more robots exhibitions than maths exhibitions :) but depending on your budget, you could find something.
- and finally, you can of course make a gift which has to do with another hobby of his. One which is more of a hobby (= he has been seen spending free time on it without any benefit beside the joy of it) than an interest (= he perks up when he hears about the matter, and maybe prefers it in school, but does not necessarily allocate free time to it).