The first production of this play was in New York City, March 11th, 1959, according to my edition. The poem that the title of this play is based on is by Langston Hughes, and captures the feelings of frustration, anger and despair of the characters in the Younger family, each of whom seem to have their dreams threatened or attacked for various reasons. Note, for example, how Beneatha reacts to her brother when he argues that her dream of becoming a doctor is rather unrealistic:
Well – I do – all right? – thank everybody! And forgive me for ever wanting to be anything at all! (Pursuing him on her knees across the floor) FORGIVE ME, FORGIVE ME, FORGIVE ME!
Her sarcastic response and the extent to which she takes it, pursuing her brother across the floor on her knees, shows her determination to succeed in her dream no matter what other people think and say. These are characters, like in the poem, who have a variety of different responses to their dreams constantly being deferred, and the title calls to question how long a dream can be deferred before it "explodes" in an act of violence.