During the time of The Importance of Being Earnest, how would the Victorians define "earnestness"?
The literal definition of the adjective "earnest," which differs from the name "Ernest" by one letter, is "Resulting from or showing intense conviction" (Google). In this context, since the word itself is less important than the name (except until the final lines) the trait and virtue of "being earnest" is one that would include honesty, dependability, and having "intense convictions" or honest opinions that do not change. When Algie finds out that Jack is not named Ernest, he is astonished:
Algernon. You answer to the name of Ernest. You look as if your name was Ernest. You are the most earnest-looking person I ever saw in my life. It is perfectly absurd your saying that your name isn’t Ernest. It’s on your cards.
(Wilde, The Importance of Being Earnest, gutenberg.org)
Despite Algie's own dishonesty, he is shocked to find that Jack is less-than-truthful; to the mind of the typical upper-class Victorian, going back on a fundamental truth (such as one's name) is more shocking than theft or murder, both of which could be excused according to rules of honor. In fact, the "importance of being earnest" is more on the level of a moral code understood implicitly by the citizens. Algie has always assumed that Jack (Ernest) was a staunchly moral person to whom the idea of dishonesty would never occur, and the revelation is both shocking and amusing.
One of the main points of the play is the satire between the name "Earnest" and the characteristic of "earnestness." I am guessing it is about sincerity. The whole play is a comedy of manners during the Victorian era. Someone earnest, in this respect, is someone with complete integrity, someone who is dependable and does the right thing, someone who is impeccably moral, and beyond reproach in any way. A confidant, a benefactor, someone who takes care of his family. Earnest in the play is a single bachelor who takes on a ward of an orphan teenage girl to raise as his own daughter because he is a man of means. He takes it upon himself to basically sponsor a young orphan girl because a woman with no family or history would have no chance at as self-sufficiency or survival without a wealthy benefactor of class. So, someone who is earnest is generous, benevolent, and just a steadfast friend, 100% dependable.