Where I live, there's a (retired) opera singer who goes by the name
Mimi. Her daughter is
Mia. The 'real' name both of them would have had, or perhaps they actually have it, would be
Maria. However, there's nothing in the rules against naming someone with a nickname form; there's gazillions of women called
Elise and
Gretchen, and men called
Don, to prove it.
In the case of
Mia, my only hesitation would be the Italian connection, with
Mia meaning: mine, belonging to me. That seems less fun than being a secret
Maria! But, my son had a girlfriend once whose only name was
Mia, and she was English-speaking with no links to the name
Maria or the Italian language. Her parents liked the name, used it, and the thought police still haven't found them.
All the best