What is the importance of being Earnest in the play?
In Oscar's personal world, there is absolutely NO importance of that. In the world in which Oscar lived, however, it was the current meme. "Being Earnest" as a meme suggests the quality of life that Victorians wanted to achieve or to at least pretend to achieve in front of their peers. Wholesome, true, religious, just think "Waltons" and you'll be good to go.
Oscar made this play seem so obviously punny that if any "true" Victorian at the time actually laughed during the play, he or she must have known that they were actually laughing at themselves.
Hence, the importance of "being Earnest" is that it embodies the values of the time in a society where hypocrisy was King. (or Queen?)
It is a pun. The play is all about two things: 1) The importance of being Earnest, the person (in order to find love), and 2) The importance of being earnest, in the sense of being sober, honest, straightforward. This was a major social concern at the time, and Wilde stands it on its head, making fun of it as a role that people play.