The house represents the hope for a positive change, a hope which will keep the family together.
Initially Mama and Ruth are the only two characters that see the house this positive way. Walter and Beneatha do not prefer the house as means of improvement for the life of the family early in the play. Later, they come to see the house as Mama and Ruth do.
The house that Mama buys stands in direct contrast to the small apartment where the family has lived for so long. Cramped, dark, and worn, the apartment symbolizes the family's history of material struggles. The new house, oppositely, represents new possibillities, a literal expansion of space.
Additionally, the new house represents a significant risk. Mama is taking a chance on her family by putting a down-payment on the house, calculating that an improvement in conditions can lead to an improvement in relations.
It is apparent it is not going to be easy for them; it might be very dangerous. But the people are transformed, and not afraid.
She hopes the diginity of home ownership and the physical improvement of conditions will make life better for the whole family and help each of them to find value in themselves.