In the South there is a custom of addressing people with their first names. Usually, when a nickname is becoming of a person's characteristics they go for a nickname that sort of identifies them. 

With kids it is not an exception. We know from the beginning of the story that Scout's name is Jean Louise and this is hardly a name that would fit her. She even admits to hating it herself.

Jem's real  name is Jeremy and for a child that is a bit long and serious to be addressed, so they shortened to Jem .

However, it would be unfair to say that most of the characters have nicknames because neither Tom, Mayella, Mr. Ewell, Calpurnia, nor the aunt are addressed by nicknames, but respectfully by their respective names.

Hence we could safely assume that the practice is one specifically given to children to shorten their names and allow them a level of individuality.