In A Raisin in the Sun, what does the house symbolize that the Younger family hope to move into?
This question can be answered in a number of ways. First of all, consider the way in which their present abode is compared with the house that they hope to have. The place they live in at the time of the play is small, cramped and infested by insects. It has no yard. Mama's plant, which is another object with massive symbolism in the play, is not thriving in this home, because it does not have enough light and space to thrive.
By contrast, the house that they hope to move into has more space, and has a yard, which is particularly important for Mama, as she is the nurturer of the family as its matriarch. In particular, her plant symbolises her desire to see her family live in an environment where they can thrive, rather than just survive. So, in a sense, the house symbolises a fruitful future for the Younger family.
However, it also symbolises something else, too. Consider how Walter refuses the offer made by Mr. Lindner on behalf of the community where they are about to move:
We have decided to move into our house because my father—my father—he earned it for us brick by brick. We don’t want to make no trouble for nobody or fight no causes, and we will try to be good neighbors. And that’s all we got to say about that. We don’t want your money.
Therefore the house also symbolises the hard work of Walter's father and his hope of having a home for his family. Because this money has been gained through the death of Walter's father, and it has been earned through years of hard, solid labour, the house symbolises a right that the Younger family fully deserve.