What is the name of Atticus's wife, who died when Scout was two, in To Kill a Mockingbird?

The first name of Scout's mother is not revealed in To Kill a Mockingbird.  Her maiden name was Graham before she married Atticus and became a Finch.  Very little information is given about her in the story.  The primary reason for this is because the narrator, Scout, has no memories of her mother.  Scout had been only two when her mother passed away from a heart attack.  Her heart condition was hereditary.  Her young age caused Scout to later say that she "never felt her absence" (Chapter 1).  

When Scout's mother and Atticus got married, he was a middle-aged man.  She was fifteen years younger than him.  She was from Montgomery, Alabama, the state capital.  Scout does not miss her mother like Jem does.  Sometimes he remembers her and wants to spend some time alone.  Uncle Jack once tells Scout that she resembles her father more than she does her mother.  Mrs. Dubose recalls Scout's mother as being a lovely person.  Scout also hints at what her mother had looked like:

Jem's soft brown hair and eyes, his oval face and snug-fitting ears were our mother's... (Chapter 15).

What is the name of Atticus's wife, who died when Scout was two, in To Kill a Mockingbird?

We never learn the first name of Atticus' wife, Scout's mother. What Scout does reveal is this in the firs chapter after she tells us all about Calpurnia's presence in their home:

Our mother died when I was two, so I never felt her absence. She was a Graham from Montgomery; Atticus met her when he was first elected to the state legislature.

Heritage was very important in Maycomb County. You will find as the book continues that Aunt Alexandra feels that Jem and Scout were carefully bred by a process called gentle breeding. This means since Atticus was a well-to-do man, his wife was from a high class family too, they were from Montgomery and her last name was Graham.

I think Lee never reveals a first name for a couple of reasons. First, the death of the mother is a fact, a detail and a circumstance of life. I don't think Lee wanted another persona hanging around the storyline. Jem thinks of her enough. Also, having a greater identity of this dead woman would certainly bring her into the story among the women of the town and certainly Scout. To have explained her anymore would be more detail than this already jam-packed novel could handle.

When is Atticus Finch's wife mentioned in To Kill a Mockingbird?

Scout mentions her mother in the first chapter. Since she died when Scout was two years old, Scout doesn’t remember her and doesn’t really miss her. Jem, who is older than Scout, does remember her. 

She was a Graham from Montgomery; Atticus met her when he was first elected to the state legislature. He was middle-aged then, she was fifteen years his junior. Jem was the product of their first year of marriage; four years later I was born, and two years later our mother died from a sudden heart attack (Chapter 1). 

As Scout grows up, it is often mentioned that Atticus is raising his children without a woman’s influence. We get a good idea of the role Scout’s mother was supposed to have played in her life, but not necessarily the kind of person she was. Scout’s Uncle Jack tells Scout she is more like Atticus than her mother (Chapter 9). 

Mrs. Dubose tells Scout and Jem it’s a pity their father never remarried, and that she liked their mother. 

A lovelier lady than our mother never lived, she said, and it was heartbreaking the way Atticus Finch let her children run wild. I did not remember our mother, but Jem did—he would tell me about her sometimes—and he went livid when Mrs. Dubose shot us this message (Chapter 11).

With his wife no longer there, Atticus serves as the sole parent to Scout and Jem. When Aunt Alexandra complains about Scout wearing overalls, he says he does the best he can. Atticus raises his children to value what he values. He wants them to be good people. It is more important to him that they respect everyone, regardless of who the person is, than that they understand their place in the social hierarchy of Maycomb. 

Aunt Alexandra comes to live with Atticus during the trial to offer support, and to provide Scout with a female role model. Although she and Scout do not get along, living with Alexandra does change Scout’s perspective on what it means to be a lady. She comes to respect her aunt and even model her behavior on hers in some ways.