When Atticus treats Calpurnia, a black servant, like a family member and gives her rights in To Kill a Mockingbird, what does this show about his character?
Just to clarify, "rights" in the question refers to allowing Calpurnia to educate and discipline the children.
Atticus Finch is colorblind when it comes to the races, and he not only welcomes Calpurnia into his home but also allows her far more responsibilities than the average African American housekeeper of the 1930s. Atticus makes it clear that Cal, who previously worked for Atticus's father at Finch's Landing, is more than just a servant.
"... I couldn't have got along without her all these years. She's a faithful member of this family..." (Chapter 14)
Cal moved with the Finches to Maycomb after the death of Atticus's wife, and she immediately became both a housekeeper and surrogate mother to Jem and Scout. Cal had learned to read and write--a rarity for Southern Negroes of the time period--at Finch's Landing, and she also assisted in the children's education. Because of Atticus's stature in the Maycomb community, Cal was elevated to a prominent status among the black members of the Quarters. Atticus gave her a free rein with his children when he was away from home, and Jem and Scout knew better than to try and take advantage of her.
Our battles were epic and one-sided. Calpurnia always won... and I had felt her tyrannical presence as long as I could remember. (Chapter 1)
Atticus apparently had no plans to remarry, and he had complete trust in Cal's ability to serve as the female head of the household. Cal would have to take a back seat once Atticus's sister, Alexandra, came to live with the family, but Atticus put his foot down when Alexandra demands that Cal be fired.
"... you'll simply have to accept things the way they are. We still need Cal as much as we ever did." (Chapter 14)
Atticus knows that no one else--black or white--could have raised his children any better than Cal.
"Besides, I don't think the children have suffered one bit from her having brought them up. If anything, she's been harder on them in some ways than a mother would have been... she's never let them get away with anything, she's never indulged them the way most colored nurses do. She tried to bring them up according to her lights, and Cal's lights are pretty good--and another thing, the children love her." (Chapter 14)