Found mistake in my professors book: Tell her or publish? If so, how?

First: Slow down.

In the academic world, books are simply collections of known things presented in a unified fashion. In fact, in some disciplines, it's almost unheard of for someone to release something novel in book form. Additionally, while book chapters are reviewed by editors and (occasionally) peers, it's not the same peer-review setting as we have in journals. To that extent, people don't typically write papers rebutting an interpretation put forth in a book; they write papers rebutting the article(s) upon which the book chapter is founded. If a book chapter does indeed contain an error, then it'll just have to be fixed in the next revision, if there is one. As such, your concern about publication is probably far overstated.

Secondly: Are you sure you have found an error? Has anyone else reviewed your work to validate what you've seen? Are you sure that you fully understand the topic? If I was a betting man, I'd place my money on the professor who's been studying the field for years and has (presumably) thoroughly reviewed the literature, so much so that they just wrote a book on it, over first- or second-year graduate student, as a simple matter of statistics.

Thirdly: You should definitely just go talk to your professor and ask her, respectfully, to clarify what she wrote. If anything, she'll probably be appreciative that you're going through it in such detail. If you have found a real issue, you'll make her aware of it, and if you're lacking some background she can explain it.

No, you won't make an impression by being bold. You are a Master student and she is a Professor, even if you are 100% sure of what you say, you can't just throw your arguments at her face: because you have to show her respect, even if what she wrote is blatantly wrong.

The best way is to point it out as a clarification question, like:

I don't understand this point, because it seems to contradict with [what I have read in another book ; what we know about the organization of this particular structure ; with the latest research ; etc.].

In addition to not being bold, this approach has the benefit of allowing you to present your arguments point by point without seeming like you are arguing (you are trying to clarify a point you don't understand, so you can continue until you clarify it enough). And if it's your mistake, then you will learn something and you can go away without being tagged as a "smart-ass".

I used this approach on several occasions, and it always worked wonderfully well, and even opened a few opportunities (because it looks like you are pretty interested in the subject, which is probably true since you could spot an error).

And remember that making mistakes is easy, particularly when writing a book! So don't be too harsh on someone just because a few errors slipped in: this is bound to happen. However, if the book is full of errors, then you have another problem: do you really want to work with someone that is clearly incompetent or delusional? In this case, you don't need to point any error, you should just get out as fast as possible!

/EDIT: after OP comment clarifying that it is about a substantial reasoning error in a theory and not just a factual error: you can of course discuss with the professor, but be prepared that the debate can become heated. And that's totally normal and you should understand it: she studied the field for years, spent a couple more years to design her theories, and you think you can break her theory at your level of knowledge and study. Of course, this is possible: it is always easier to find a counter-example than to define a new theory. But remember that she put a lot of work in her theory, so even if the theory is wrong, you have to respect her for her effort in trying to advance the research in her domain.

Now you are free to either talk to her or publish a paper, it's your choice, but be sure to respect the person behind the research: as a researcher, your goal is to rebuke/confirm hypotheses and theories, not attack the person designing them. Focus on the content rather than the person (forget about trying to impress), and you should be good.

The simple answer is that you should do both.

I do not believe you owe a professor just because she is a professor. If she has earned your respect and/or your loyalty through other means, you should certainly demonstrate it, but being a professor doesn't demand it, on its own.

That said, talking through the problem you have found with her will give you a lot of clarity on whether it is the scale of error you believe, and will help sharpen your own argument. It may not be an error at all in your professors mind. I doubt she'll hear your reasoning and slap her head, exclaiming "Oh snap! I was wrong all along!" She may simply believe it is unimportant, which is fine. It then falls to you to justify the importance of your observation.

Then you should publish this idea. The object of academia is to engage in the "Great Debate" and if you have found a doorway into that debate, and you believe in your argument, then you should absolutely walk through that door. If your professor has published the opposite, then all the better because it is something that is already in the "academic consciousness" and will consequently be relevant to both the discipline—assuming your professor is not a marginal figure—and your immediate environment.

I don't understand where the idea comes from among students that most good ideas are 'sprung' on the world through publication, or are kept private until the last minute. I know this is a common idea but the reality is much more complex, and really doesn't vary that much by discipline. Most ideas are talked out, written, then rewritten, sent to conferences, sent to journals and finally published, and that process does and should include the hostile interlocutors. By the time most ideas get published, regardless of article or book, the idea has already been widely discussed. Publication isn't really the revelation of an idea to the community but the codification of an idea, and an attempt to spread it to a wider audience.