Honesty is a form of ethical and moral behavior, and in Harper Lee's To Kill a Mockingbird, Atticus' moral compass is what drives his behavior.
The most common things we see in Atticus' behavior are caring for those who cannot care for themselves.
Shoot all the blue jays you want, if you can hit 'em, but remember, it's a sin to kill a mockingbird.
This includes not only mockingbirds, but also those characters symbolic of mockingbirds, including Boo Radley and Tom Robinson.
However, the segment of the story where the reader so clearly sees Atticus' imperative to be honest is after Ewell is killed when attacking Scout and Jem. Atticus believes that Jem did killed the man in self-defense; Atticus also feels that Heck Tate, the sheriff, is trying to protect Jem. Atticus is appreciative of Tate's gesture, but he cannot turn his back on what he believes and all that he has tried to teach his children. Why would they choose to be honest if he isn't honest? He says to Heck Tate:
Thank you from the bottom of my heart, but I don't want my boy starting out with something like this over his head. Best way to clear the air is to have it all out in the open. Let the county come and bring sandwiches. I don't want him growing up with a whisper about him, I don't want anybody saying, 'Jem Finch... his daddy paid a mint to get him out of that.' Sooner we get this over with the better.
For honesty's sake, Atticus is willing to fight the entire community to defend his son's right to kill the murderous Bob Ewell. (Ironically, however, it is not Jem that Tate is concerned about, but Boo Radley, who saved the children from Ewell. A strongly ethical man himself, Tate simply wants to protect Boo from the attention such a heroic act would garner—bringing the painfully introverted Boo into the limelight of the community.)
Atticus is a man of outstanding character with a strong sense of morals and ethics. For Atticus, honesty is the measure of a person's integrity. This is in keeping with all the other things Atticus teaches the children about being kind to others and thoughtful of their circumstances, doing no harm and choosing to be honest.