I was surprised to read in The Times that Mona (as in Mona Lisa) is a common Italian short-form of Madonna, meaning My lady.
Except that a quick bit of Google research brings up many other explanations & origins – Wikipedia has a nice summary table.
That set me wondering about Monica. No relation, apparently, though she was St Augustine’s mother & is the patron saint of married women.
Moniker, an English slang term for any name, is of uncertain origin.
If you are a mathematician you will of course be familiar with the idea of a monic polynomial.
You should be particularly cautious in using the name Mona at a football ground, especially if you are the referee. According to the OED it is the name of a kind of monkey with a bluish-grey face and a pink muzzle.
Links
Wikipedia: Mona (name)
Wikipedia: Monica (given name)
Related post
Madam Madonna
Football goes bananas