The title has a dual meaning as it pertains to the play as a whole.
First, the title's significance lies in the denotation of the word "earnest." Defined as a quality of being sincere, the word is an important thematic idea in the play as a whole, since several characters use secrecy and deception in order to pursue their own interests. The tangle of lies causes several conflicts among the characters, chiefly relating to their romantic pursuit of one another. The title of the play is a tongue-in-cheek reference to the lack of honesty within the characters in the text.
The second way the title is significant relates to the homophonic male name Ernest. Both Jack Worthing and Algernon Moncrieff use the pseudonym Ernest to woo the ladies in whom they are romantically interested, Gwendolen and Cecily, respectively. For both of these women, the name Ernest is what makes the man in her life especially attractive, as she insists that she is destined to marry a man bearing the moniker.
In fact, Gwendolen nearly breaks off her engagement to Worthing because he reveals that his name is, in fact, not Ernest. It is revealed, however, that Worthing was indeed christened under the name Ernest at his birth prior to being lost and adopted by another family. Gwendolen then agrees to marry Worthing as planned. This shows that the play's title also refers to the name Ernest, since it is essential to resolving the conflicts of the play.