The Finnish author Tove Marika Jansson (1914 - 2001) invented this name for her stories about the Mumin troll (Moomin Trolls).
Lilla My (Little My) lives with the Mumin family. She is so small that she can sit in the family’s milk jug. My gives expression to a most destructive turn of mind. She is totally disrespectful and can be very aggressive but has nevertheless an extremely positive attitude to life. Sometimes, she may even demonstrate a kind of careless friendliness - when it suits her.
My is also the name of the 12th letter in the Greek alphabeth.
Apparently. Finns are and odd bunch.
See: https://www.nordicnames.de/namefinder/