Indicates an admission of defeat in a battle of wits. The verbal equivalent of waving a white flag.
Alternatively, it is used in an act of petulant defiance, similar to the way children shout and stomp their feet when they don't get their way.
Person 1: I am going to the doctor's office today. I have to see a specialist.
Person 2: Veterinarian?
Person 1: Meehn.
Person 1: Stop ignoring me! Pay attention to me!
Person 2: I'm sorry, do I know you?
Person 1: MEEHN!