The house does not mean the same thing to each member of the Younger family initially. However, by the close of the play the house comes to have a single meaning for the whole family, ultimately representing a chance to come together as a family.
Walter and Beneatha are both more interested in achieving individual success and in finding some personal dignity through individual achievement. They each feel that looking outward for success is the surest way to build up a pride. They do not look at family as being a source of pride.
This outlook carries over to the views that Walter and Beneatha initially hold regarding the prospect of buying a house. As the house is meant for the whole family, it does not seem to satisfy the individual needs of these two characters.
For Walter, in particular, the house represents an alternative to his plan to buy a liquor store. By extension, the idea of buying a house serves to undermine his authority in the household. It is not what he wants to do. The house offers no possibility for moving up in the world financially. The house will not change Walter's position in the world as he sees it.
For Mama and Ruth, who see the family falling apart, the house does represent a kind of possibility. The apartment is worn out, small and dark. The family is following the same arc as the apartment.
"Its furnishings are typical and undistinguished and their primary feature now is that they have clearly had to accommodate the living of too many people for too many years."
A larger house will offer comfort to the family and may ease some of the inevitable tensions of living in a cramped space. Additionally, the new house will take the family out of the impoverished environs where they currently live. The house represents a chance for the family to remain intact.
When the family's pride is put on the line, both Beneatha and Walter begin to see the house as a means to achieving dignity. Choosing to move into the house will not only please their mother. It will demonstrate their own view of who they are and what value they hold as people and as a family.
Walter is able to prove that he has pride and dignity by refusing to be kept out of the new neighborhood. This course of action also allows Walter to choose to define himself through his family. The same is true of Beneatha.
The house, finally, becomes a symbol of hope and faith in the family structure, in the unity of the Younger family, and in the potential of the family unit to be a generative source of individual value and positive identity. Mama had worried that Walter would not be able to find this value and this pride in his family.
As the play ends, Mama talks to Ruth about the change in Walter that happened that day; both women are very proud.