What do the children represent in "The Lottery" by Shirley Jackson?

This is a very open-ended question, so you approach it from a number of angles. There are a few tried and true attributes that children have that are frequently represented in literature. One example of this is childhood innocence. Everyone's world view is limited by their experience, and the experience of a child is naturally limited due to their young age. So, they often see things in an idyllic light.
William Blake had an underlying theme of this in many of his poems. His theme was that people pass through an initial age of innocence, and then eventually enter an age of experience. The children in "The Lottery" are still in the innocence phase of life, since they have not experienced the good and bad, the triumph and the struggle that their parents and grandparents have so far. This is notable because Shirley Jackson could be using the children as a gauge for what is about to unfold in her story. Readers can understand the perspective of a child and instinctively know how children think, thus the reader can form an opinion of the the lottery happening in the story. 
Children are also used as a means of further contrasting innocence and experience. At one point Jackson shows a young child (two years old) as the one to draw the slip with the name of the person determined to die. But, at the same time, she also shows older boys who have learned what this means. They are gathering rocks, ready to pelt the chosen person to death. These boys have begun their transition from the age of innocence to the age of experience. 

The children are probably intended to represent the next generation of citizens who are being taught how to participate in the annual lottery. It is because each new generation of children is taught to participate, and sees their parents and parents' parents participate, that the lottery continues. The whole procedure of drawing and stoning is solemn and serious, although there is no sense to it, and never was. If children were not indoctrinated at the earliest age, the lottery would cease to exist. Little Davy, who is a member of the Hutchinson family, appears to be only about two years old, yet he is being shown how to draw the slip of paper which could contain the black spot that would mean his own death. A group of boys who are a bit older already know the important rules. They are collecting rocks to throw at the person who will be selected. All ages are represented in the story for the purpose of showing that the lottery is an age-old tradition. Davy will be like Old Man Warner someday, if he manages to survive all the annual drawings. By that time, Davy will be thoroughly convinced of the value and necessity of this superstitious event.