Certainly the purpose lies first ans foremost in the purpose of the play as a comedy of trivialities. This being said, Oscar Wilde intended to make this an ironic, pun-filled, and sarcastic comedy of manners in which the upper classes and their hypocritical behaviors would be the motif of ridicule. He chose the name Ernest as a pun for earnest, and the ladies Cecily and Gwendolyn as a representation of the superficiality of the marital state. Hence, with both women falling in love with the name of Ernest, and he being just called Ernest as a fake, exposes him as a liar, the opposite of an earnest man.
The name thing was clarified in the end, when Jack finds out that his father's name was Ernest, hence he is an Ernest "by default". Still, Jack has the nerve to say at the end "I finally found the real importance of being earnest". Basically he is exculpating himself from the whole double personality deal! In there comes the triviality of the play, the irony behind the name, and the reason for it being the central focus of the title.
In my point of view the purpose or the significance of the title is this "the word Earnest has great importance in the lives of characters especially in Gwendolen and Cecily" both these girls think and express their feelings regarding this name before their beloveds that this name has music in its own. Jack Worthing and Algernon Moncrieff both assume this name before their beloveds and their family.
Another aspect of the significance of the title is this " this play Importance Of Being Earnest is a satire and Oscar Wild satirizes the foolies of characters and he wants to say that a name has great importance and passions and emotions are meaningless. Gwendolen and Cecily both only give importance to the name "Earnest" and they do not admire and appreciate and accept their love and their emotions and their feelings.