The name Hermia is the feminine form of Hermes. Hermes was a Greek god who was the messenger of Zeus. He was also assciated with speed and good luck. Hermes was also the patron of travellers, writers, athletes, merchants, thieves and orators.
A reason why Shakespeare may have chosen the name Hermia might have been to display her frustrations and needs during The Midsummer Night's Dream. Hermia needed speed and good luck to help herself and Lysander to escape. Also, there were miscommunications between Hermia and Lysander after the flower's sap was applied to his eyes. The message Hermia got from Lysander was that he no longer loved Hermia, but he loved Helena instead. But, as we know, Lysander was bewitched by the flower's magic powers, and things were resolved in the end by the ever so mischievious Puck.