As a PhD student, how do I benefit from reviewing a book my adviser is writing?

You'll gain his trust, if you do a good job. You will also gain a deeper understanding of the material as you'll read it thoroughly not just to learn it, but also having to think about how the material is presented. Don't expect anything else.

In questions like that, it is a good exercise to play it through being contrarian, e.g. as follows: "I do not get any rewards for helping my adviser. I am not going to tell him that there are errors in the problems when I encounter them - let him sort that out himself, it's not my business. And if I were to do it, I should mention that I expect to be named as at least sub-author, or prominently in the acknowledgements, or I certainly won't do it."

This is (hopefully) slightly exaggerated, but do play it through - and now adopt your supervisor's perspective in how your advising relation will look like if he sees your view this way and decides to treat your advisory relation in the same fashion.

Importantly, note that you TA the course anyway, and will encounter flaws in the exercises, so it is not likely to be undue extra work to help him find errors in the book.

It would be a different story if you had a strained/unhelpful relation with your advisor, but you mentioned nothing in the question to indicate you have.

Bottom line: you may go for a minimal effort in helping your adviser - but do not expect more than a minimal effort in return then.

It should be obvious, but I will explain this nevertheless. Reasons for:

• You are TA for the course. Who is the second most appropriate person to review the book for this particular course? First one is the professor, well, the second one is you.
• Poorly written book will be your problem too, as students may expect you to explain things that are not properly explained in the book. So it is in your best interest to help the professor produce a good textbook, it will make your life easier too.
• Undoubtedly the professor trusts you enough to ask for your feedback. Be nice and show him that he is right in trusting you.

Now, possible reasons against:

• You are not really into your TA job and this is only additional burden for you.
• You are not confident you master the field enough to be TA, so you are not confident you should give feedback on the book.

If any of two latter are the case, you are right not to bother with the book, but on the other hand, you should think really really hard whether you want to be TA on this course. I do assume that the rejection of the offer to review a book will be interpreted in this sense by the professor as well.