What does "worrying another bone" mean in Chapter 9 of To Kill a Mockingbird?

"Worrying another bone" in Chapter 9 means to fret about another matter. In Chapter 9, Scout is speaking about how she doesn't want to go to school. She says that "the beginning of last September had brought on sinking spells, dizziness, and mild gastric complaints." She will do anything to avoid going to school, and she tells Atticus she shouldn't go to school if she is going to learn common words there. Her first concern is trying to avoid school by any means.

Then, she starts worrying another bone, meaning she begins to think about other matters that concern her, and she starts asking Atticus whether all lawyers defend African-American people, as Atticus is defending Tom Robinson. Atticus responds that if he didn't defend Tom Robinson, "I couldn’t hold up my head in town," meaning that he couldn't act with honor or believe that he had a sense of integrity if he didn't take Tom's case. Scout's worries cause her to ask a series of questions through which she learns a great deal about Tom Robinson's right to a trial and about Atticus's commitment to the case. 

In chapter nine of To Kill A Mockingbird, Scout is speaking with Atticus.

But I was worrying another bone. Do all lawyers defend n-Negroes, Atticus?

Scout is worried about her father's reputation, even though she is young. When others accuse Atticus of being a  "nigger-lover," Scout is ready to punch them with her fist. Even though she is not quite old enough to understand, she realizes it is a negative insult.

It is brilliant that Harper Lee uses the innocence of an eight-year-old girl to ask questions that really shine the light on the prejudices found in Maycomb. Scout exposes people for what they really are--prejudiced. She understands that their prejudices are hateful and should not be allowed to exist. Scout is offended by people's prejudices. She knows that it is wrong to be prejudiced.   

What is the meaning of the idiom "worrying a bone" in Chapter 9 of To Kill a Mockingbird?

This passage comes in Chapter 9 of To Kill a Mockingbird when Scout is once again trying to convince Atticus that she doesn't need to attend school. But Scout realizes that debating the subject with her father is like "worrying another bone." This is an old phrase that literally means

"To seize with the teeth and shake or tug at repeatedly: a dog worrying a bone..."

Rhetorically speaking, however, it reflects on the argument that Scout and Atticus have had before about school--a verbal tug-of-war and back-and-forth debate that Scout knows she will not win. The idiom is similar to the one about "flogging a dead horse"--it's a useless action with predictable results. Thus, her statement about the "bone," followed by her changing the subject to that of lawyers defending Negroes.