Events prior to the manga
Hana's wings were plucked by goblins as punishment for minor theft. As a result, she is bad tempered and abrasive, though how much so depends on whether all fairies are mean, such as the one that bit Sarah in the movie.
Role in the manga
She befriends Toby during his journey into the Labyrinth and he promises to help her find her wings. When Toby's studies prevent him from helping her search for her wings, she become agitated with him and goes with Moppet to find them alone.
In a way, her friendship with Moppet makes a lot of sense. Both were considered outsiders by goblins. Hana is just some vermin while Moppet is a poser who only exists because the King wanted it. Hana's wings make her feel incomplete, while Moppet is literally incomplete, an incomplete soul.
If the manga continues the implication that the various friends met along the way are psychological interpretations, Hana would be Toby's snarkiness, his attempt to have a rebellious sense of humor to get through dark times.
Hana has a fierce temper and is easily agitated.