The title refers to the line at the end of the play spoken by Pegeen, "I've lost the only playboy of the Western World!" She says this after Christy reconciles with his father and decides to travel with him. Since Christy has decided to better himself and no longer is interested in Pegeen, she is morning the loss of his novelty, as he is a "playboy" no more.
It should be noted that in the time and place of the story, the term "playboy" does not imply the modern association with the word. When the characters of the play refer to a Christy as a "playboy," they are calling him out for being a trickster and lying about the murder of his father. The term can be slightly endearing, as Christy's talent for talking himself up and storytelling can be praised.