This analysis website claims that:
In Latin, Atticus is an adjective meaning “belonging to Attica”, the region in which Athens is located, or more simply, “Athenian”. As a name, it had connotations of literary sophistication and culture.
Atticus was a suave and charming wheeler-dealer who deliberately eschewed political office.
He preferred to exert influence through his business interests and by cultivating powerful friendships.
Although rare in the Republican period, the name Atticus became much more common under the Roman empire. The evidence of epitaphs and other inscriptions shows that it was held by senators, centurions, cavalrymen, town councillors, freedmen and soldiers in the praetorian guard.
But it did not lose its association with Greek culture and refinement. One of the most famous Attici was Herodes Atticus, an extremely wealthy intellectual and orator who was appointed as tutor in Greek rhetoric to the future emperor Marcus Aurelius.
This source states:
Atticus' name is a nod to the Roman orator known for his just decisions.
The Wikipedia article states that it is an "adjective Latin name meaning 'classic, elegant'".
(emphasis mine on all of the quotes above)
In essence, the bolded words represent Atticus' character very well. He is a sophisticated, highly educated, and well-cultured intellectual who always attempts to make the most ethically right decision in any situation. His name was carried by powerful Roman orators, senators, and councillors, and his occupation as a charismatic lawyer reflects this. Lee chose this name because of its historical connotation of class, elegance, sophistication, and intellectualism, all qualities that embody Atticus Finch.