The suffering child in LeGuin's story "The Ones Who Walk Away from Omelas" is the scapegoat for the misery of others, so that the others are able to live in comfort and happiness.

The concept which LeGuin explores in this work of philosophical fiction is that of the greater good, a concept explored in William James's The Moral Philosopher and the Moral Life. In this work James writes on a supposition that one person might be able to absorb the pain, illness, and misery for all others:

Or if the hypothesis were offered us of a world in which...millions [could be] kept permanently happy on the one simple condition that a certain lost soul on the far-off edge of things should lead a life of lonely torture....

This miserable creature that is confined is essential to the comfort of all the others in the community. In Omelas, then, all the "goodness and grace of every life...." depends upon the condition of this child imprisoned in a basement of one of the attractive public buildings. This miserable creature must be kept there for "the greater good" that allows others happiness and comfort.

When children reach the ages of between eight and twelve, they are brought to see this isolated creature, and all are shocked at the sight. But they are instructed that if something were done for this miserable creature, the "beauty and delight" of Omelas would wither and die. Only by exchanging all the goodness and grace of every life in Omelas can the single child's life improve. While some residents of Omelas are so disturbed by the discovery that they "walk away from Omelas," departing into the unknown, others become reconciled to the idea that one life where evil is contained is worth the happiness of all others. Further, they feel that even if the child were released, it has been degraded for so long that it "would not get much good of its freedom," anyway, because it has become "imbecile."

With this vagueness of position at the end of LeGuin's allegory, the readers must, then, draw their own conclusions of the author's particular illustration of Pragmatism.

The suffering child functions to set up the central conflict in the story. At first, Omelas sounds like a perfect utopia, a beautiful, joyous place that combines the best of Medieval community and loveliness with the best of modern technology. It is hard to imagine not wanting to live in such an ideal place.

All of that changes when we discover that the happiness of this world is based on the abuse, neglect, and suffering of an innocent child. It is morally repulsive to imagine our own happiness to depend on the continual pain and misery of a young child. We learn that the society has rationalized the child's suffering as acceptable because this suffering has enabled so much joy. The society has embraced the philosophy of Utilitarianism, which defines happiness as that which brings the greatest good to the greatest number of people. From the Utilitarian point of view, the suffering of the child, while it might make people uneasy, is worth it because of the benefits it brings to so many.

However, the awareness of the child's existence is a source of discomfort that taints and threatens to spoil all that is good. Can the culture be good if it is based on cruelty? Can any person really enjoy all the good things they have knowing that they are enabled by another's suffering?

The child represents suffering and pain. She represents what people often do to children in marginalized situations who toil so we can have nice lives. Because of the child, every citizen of Omelas and every reader has to make a decision: would they accept a child's pain to achieve their own pleasure, or would they walk away?

What role does the tormented child play in the story "The Ones Who Walk Away From Omelas"?

In "The Ones Who Walk Away From Omelas," the utopian society of Omelas is whatever the reader makes of it; it can be as urban or rural as desired. However, the entire fantasy of Omelas is dependent on the suffering of a single child, living in filth and locked in a closet. All the citizens know about it, and most accept it as the price for their happiness and prosperity.

The child is normally identified as an allegory; the people of Omelas accept its mistreatment because it allows them their own comfort. Similar situations are seen all over the world, and in most developed countries the majority of the middle-class population ignore the lower-classes, homeless, or poverty-stricken people because they feel they can do nothing substantive to help them. They are not themselves mistreating the poor, but they are not doing anything to help them either.

In the story, the child stands for the unstated moral code of Omelas: 

If the child were brought up into the sunlight out of that vile place, if it were cleaned and fed and comforted, that would be a good thing, indeed; but if it were done, in that day and hour all the prosperity and beauty and delight of Omelas would wither and be destroyed. Those are the terms.
(Le Guin, "The Ones Who Walk...," liferoar.wordpress.com)

"The Terms" are the unspoken agreement of all that the child's mistreatment is acceptable given the utopia that exists. With everything so good, why should the suffering of one single person matter? The moral code of Omelas could read: "The Collective takes precedence over the Individual." The child, who might have been entirely healthy at birth, is simply sacrificed with no thought to its future so the utopia can survive. The people of the title, who are so struck by the situation that they leave Omelas and never return, are the people who feel that they cannot save the child but will not continue to be part of the society that deliberately mistreats a single person for its own benefit.

What is the significance of the child in the short story "The ones who walk away from Omelas"?

The child in "The Ones Who Walk Away from Omelas" is significant as a scapegoat who bears the burden of misery for the community. This child also represents the dilemma of pragmatism.

With the subtitle of LeGuin's story as "Variations as a Theme by Williams James," the narrative addresses the Jamesian concept of Pragmatism. In the quasi-utopian society, then, the deprived, mean, and vulgar are accepted as a necessary part for the existence of the "greater good." This miserable child absorbs the ills of the society so that the others can live happily, so happily that "smiles have become archaic."

Those are the terms. To exchange all the goodness and grace of every life in Omelas for that single, small improvement: to throw away the happiness of thousands for the chance of happiness of one: that would be to let guilt within the walls indeed.

Pragmatism explains that the sacrifice of this one miserable creature is practical--for the greater good--as with its sacrifice the citizens of Omelas can celebrate life and not experience suffering. Truth, then, is what is useful. However, there are those who cannot feel guiltless in this state, and they are the ones who walk away. Others, although disturbed, accept the conditions because they are too comfortable to sacrifice this state for the improvement of just one creature. 

Therefore, this miserable little creature, who is imprisoned in squalor, presents the moral dilemma of the idea that if a course of action has the desired effect, it is good, and is only wrong if it is not effective.

Explain the significance of the impact that the tormented child has on the citizens in "The Ones Who Walk Away from Omelas."

In "The Ones Who Walk Away from Omelas," LeGuin attacks the philosophy of utilitarianism, which argues that society should pursue the greatest good for the greatest number.

Omelas has reduced suffering to one person, an innocent child who is kept in misery so that everyone else can live the good life. Everyone in Omelas knows about this child.

The existence of the child makes people uncomfortable and uneasy. Most in the society rationalize away the child's suffering, saying she must get used to it, so it is not so bad, or that it is not so high a price to pay for everyone else to live in comfort, or that it is wrong to get too sentimental about the child. Some people, however, refuse to live in this society, knowing that their own pleasure is the result of someone else's pain. They are the ones who walk away.

Explain the significance of the impact that the tormented child has on the citizens in "The Ones Who Walk Away from Omelas."

I think that the impact the child has on the citizens of Omelas becomes one of the defining elements of participation in the community.  In the end, the fundamental reaction that the citizens have to the tormented child determines if they remain in Omelas or walk away from it.  If the citizens are content with their happiness coming at the cost of the child, they can remain in the community.  Their participation in Omelas and the joy that comes from it is one that they can rationalize at having come at the cost of the child's suffering and state of torment.  Essentially, if the citizens of Omelas cannot accept that their happiness comes at the cost of the child's pain and suffering, they leave it.  This is significant because it shows in clear terms how remaining in the community is something that must be understood based on the reactions one has to the child.  It is here where I think that the role that the tormented child plays and its impact on the citizens bears importance.  Participation in the community is directly related to how its citizens perceive the suffering of the child as something that they find acceptable or something that is deemed unacceptable.