There are certainly many theories as to why Scout and Jem address their father as Atticus. Perhaps Atticus wants his children to see him as someone they can trust and depend on when the chips are down; if we look at the text, we see this desire clearly in the way Atticus communicates important life lessons to his children. Atticus' patience is only exceeded by his steadfast adherence to honorable conduct; while he never laughs at childish ignorance and never minces words when a situation calls for it, he does not tolerate unprincipled behavior from Scout and Jem. He is able to engage his children while maintaining clear evidence of his love for them in their communications.
His style of parenting is such that he actually takes the time to reason with and to talk with his children as if they possess both the emotional and mental intelligence to comprehend the rationale behind his actions. He treats his children with great respect; I would say that this great belief in egalitarianism is probably the main reason why he allows his children to address him as Atticus. He never belittles Scout and Jem's intelligence by exhibiting indifference, apathy, or disgust towards their immaturity. To Atticus, his children are human beings worthy of the same consideration and respect adults deserve. Both Scout and Jem can trust that Atticus is always going to be honest with them and that, no matter what happens, they can trust him to be fair.
Take for instance the situation when Jem cuts down every single one of Mrs. Dubose' camellia buds. Jem does this because of his loyalty to his father. Mrs. Dubose's ugly accusation of Atticus lawing 'for niggers and trash' is more than Jem can stand. Let's look at how Atticus deals with it:
“Son, I have no doubt that you’ve been annoyed by your contemporaries about me lawing for niggers, as you say, but to do something like this to a sick old lady is inexcusable. I strongly advise you to go down and have a talk with Mrs. Dubose,” said Atticus. “Come straight home afterward."
Notice that Atticus doesn't yell at Jem or attempt to humiliate Jem through condemnation. He does not resort to guilting Jem into remorse, either. He first starts out by expressing his understanding of Jem's actions and then proceeds to clearly state why Jem's behavior is wrong. Then, he tells Jem exactly what he needs to do to address his own actions. Atticus basically treats Jem the same way he treats the despicable Bob Ewell and the hapless Mayella in court. Atticus never neglects the facts, but his delivery is always dignified, principled, and steady, even when it has to be strongly worded.
When Scout accuses Atticus of betraying Jem, Atticus answers:
“Scout,” said Atticus, “when summer comes you’ll have to keep your head about far worse things... it’s not fair for you and Jem, I know that, but sometimes we have to make the best of things, and the way we conduct ourselves when the chips are down—well, all I can say is, when you and Jem are grown, maybe you’ll look back on this with some compassion and some feeling that I didn’t let you down. This case, Tom Robinson’s case, is something that goes to the essence of a man’s conscience—Scout, I couldn’t go to church and worship God if I didn’t try to help that man.”
Atticus' response to Scout acknowledges her concerns but also highlights the truth of what the children will face when the whole town realizes that their father will not back down from his principles. Atticus' authenticity and transparency in discourse is fully demonstrated in the implications of his children's address to him.
To Jem and Scout, the name 'Atticus' means truth, friendship, respect, trust, and honor. It is the name for 'father' and 'friend.' Atticus is willing to flout the conventions of Southern norms of respect in order to unequivocally demonstrate his love for his children. He strikes me as someone who also exhibits great compassion for the limits of one's humanity. When Jem admits that he isn't really sorry despite his apology to Mrs. Dubose, Atticus responds:
“There was no point in saying you were sorry if you aren’t,” said Atticus.
So, you can see that everything Atticus stands for is demonstrated in his daily interactions with his children and his neighbors. In allowing his children to address him by his first name, he acknowledges both his humanity (and theirs) and the principles by which he has chosen to live by and to raise his children with.