Big Walter once said that god seemed only to see fit to give black people dreams but he gave them children to help them believe that dreams could come true.
Children are a fulfillment of a hope and are themselves a hope that things can change for the better.
I'm with accessteacher on this one. The children in this play are the reason for hope. When Ruth wants to abort the child she is carrying, she has clearly lost all hope for her family's future. When Mama puts a down payment on a house, it is an expression of hope for her grandchildren. Without the children, the Youngers may have simply drifted inexorably apart and eventually exploded, as one of the lines in Langston Hughes's "Dream Deferred" suggests.
The fact that both Beneatha and Walter have different dreams is an important piece of this conversation. Walter wants something of his own where he can be his own boss -- it is a more traditional dream for young men. Beneatha wants a college education and to become a doctor. These are wildly nontraditional considering the time period -- a black, female doctor would have been a rare person in the late 1950's. Beneatha is younger and freer than Walter, who is married and has children. Both dreams are valid. Both dreams are ultimately achievable. Mama's dream is for a material measure of stability -- a house -- the children's dreams represent something even more fundamental about what people need in order to define themselves in the world.
I would want to comment upon this question somewhat indirectly by focusing on one of the major symbols in this excellent play, which is Mama's plant. Mama is constantly taking care of the plant she has in their room, and admits that this plant does not receive sufficient water or light for its needs. However, crucially, it still flourishes, in spite of its lack of these things. Mama's care is shown to enable the plant to flourish. The parallel between the plant and the children in the play is obvious: Mama cares for them in just the same way, even though it is woefully evident that they are all lacking so much and are being brought up in an environment which makes it difficult for them to flourish.
The children in the play seem to function by making us think about hope and the desire for a better future, that Mama wants so she can plant this plant of hers and let it grow in a more suitable environment. Thus when Ruth talks about getting an abortion, Mama knows that she has to act, even though it might mean upsetting her son. Note what she says:
I--I just seen my family falling apart today... just falling to pieces in front of my eyes... We couldn't of gone on like we was today. We was going backwards 'stead of forwards--talking 'bout killing babies and wishing each other was dead... When it gets like that in life--you just got to do something different, push on out and do something bigger...
Hope is explicitly linked with the children in this play, and Ruth's talk of getting an abortion signals the absolute loss of hope. Mama's act in buying a house signals the return of hope to the family, as they can all begin to look forward to a brighter future where they, like Mama's plant, can finally receive the kind of environment they deserve in which to grow and flourish.
To a large extent, the children in the play help to bring some level of hope for the future. Like everything else in the play, this is shrouded in complexity and intricacy. Yet, it is there in terms of how children can represent redemption. For example, Beneatha represents a sense of hope in the future because of her freedom, her ability to exercise autonomy, as well as her sense of voice. The complex element is that there are times when her voice reflects selfishness as well as an inability to follow through on commitment. Travis represents that sense of hope in that he is able to get an education and presumably do more than his father. However, when Travis is trapping the rat, it speaks symbolically to how he, himself, might be trapped, and being unable to fully articulate his own condition, it reflects this level of challenge. Even Walter represents a sense of hope shrouded in complexity. He has the ability to act in the right way for his family, but it is deferred by obstacle after obstacle. In the end, Walter, Mama's child, does act in the right and in recognition for what is right. Like the unborn child that is inside Ruth, there are challenges and moments where doubt settle in, but the children in the play end up representing the hope and redemption that is needed to blight the despair and pain that accompanies the Younger family and those like them.