In To Kill A Mockingbird, how does Harper Lee show why individuals sometimes feel the need to challenge their society?

People don't feel the need to challenge their society when they feel like everything is going alright. If people feel they are being treated fairly, if they feel safe and protected, and have the basic needs to sustain life, then they won't protest or cause problems. However, if the opposite is true, and people are pushed to a point where they can't seem to find a way to solve problems through normal processes, they may protest or take things into their own hands.

Jem challenges the rules of being a gentlemen in society by taking out his frustration on Mrs. Dubose's camellia bushes. Scout explains what she figures drove him to it, as follows:

"I sometimes wondered exactly what made Jem do it, what made him break the bonds of 'You just be a gentleman, son'. . . At the time, however, I thought the only explanation for what he did was that for a few minutes he simply went mad" (102).

Mrs. Dubose wasn't treating Atticus fairly with her words around the children. Day after day she would unfairly call their father mean names and tell the children how horrible they are. Jem was told by Atticus to remain a gentleman at all times, even when she said her worst. But there came a time when Jem couldn't take it any longer, so he acted out in a fury and the result was a mad protest. 

Next, Atticus tells his brother Jack how he will challenge his society by saying the following:

"Before I'm through, I intend to jar the jury a bit--I think we'll have a reasonable chance on appeal, though. . . Why reasonable people go stark raving mad when anything involving a Negro comes up, is something I don't pretend to understand" (88).

Two things are happening in the above quote. First, Atticus will challenge the jury to think about their society differently by "jarring" them. But then he also says that people challenge society whenever anything involving an African American comes up. Atticus wants to challenge the jury to think more profoundly about the case at hand; whereas, "people" in his society challenge what they fear most--losing control of their society or way of life to African Americans. Therefore, some people challenge society in an effort to change it and others challenge it in an effort to keep it the way it is. 

Finally, Mayella challenges her society when she's backed into a corner and doesn't know what else to do, so she uses her white entitlement to hurt Tom:

"That ni**er yonder took advantage of me an' if you fine fancy gentlemen don't wanta do nothin' about it then you're all yellow stinkin' cowards, stinkin' cowards, the lot of you" (188).

Mayella can be chunked into the group of "people" Atticus is talking about. She fears for herself (her ego) and preserving her way of life more than she does the life of Tom Robinson, so she challenges the jury by calling them cowards if they don't side with her. It would seem then, that people challenge their society for the following reasons: when there is a noble cause to defend; when people feel frustrated with how they are being treated; when they want to preserve their way of life; or when they fear losing control of what they have.