Like Gwendolen, Cecily is attracted to the name "Ernest" because she believes it conveys a sense of reliability, honesty, and romantic idealism.
When Algernon (pretending to be Ernest) proposes to her, Cecily explains:
"It had always been a girlish dream of mine to love someone whose name was Ernest."
Just like Gwendolen, she has idealized the name, associating it with a man of strong character and deep passion. The irony, of course, is that both Jack and Algernon are anything but "earnest" (honest and sincere)—they are both deceiving the women to maintain their double lives.
This obsession with the superficial importance of a name is one of Wilde’s key satirical points, poking fun at Victorian society’s tendency to prioritize appearance over reality.