Should I mention a mistake made by a third person?

It was your task to ensure the book gets returned. You need to take responsibility and apologize. You can mention that you had made arrangements, which failed unfortunately.

It's unlikely that the professor cares about the details. What he cares about is that he didn't get the book (that's why you apologize) and that he can rely on and trust you (that's why you promise it doesn't happen again and make sure it doesn't). This is a learning experience, i.e., you know now that your relative is not reliable and as a consequence your professor trusts you slightly less (but that's not permanent damage). Obviously, make sure he gets the book as soon as possible.

I disagree with the accepted answer. Yes, it was your task to return the book so, yes, you need to take responsibility and apologize.

However, you should also mention that you delegated the task to somebody else and they have let you down. Suppose you don't mention this, your relative doesn't send the book for another week and the professor sends you another mail. How are you going to explain that you still haven't done what you said you'd do?

And suppose that you later want to borrow another book. If the professor thinks it's completely your fault that the first book was late, they'll be reluctant to lend you another. But, "I'll never get that person to return another book for me!" is a concrete and believable plan for getting the next book back on time.