Although databases are probably the most practical solution, if you want to have some fun maybe you could try writing a neural net (or using a neural net library) that takes in the name and outputs one of those 3 options (F,M,A).
You could train it using the datasets that exist in the databases suggested by other answers, as well as with any other data you have.
This solution would allow you to handle names not specifically categorised previously, and also handle different languages. You might want to pass the language (if you know it) as an input to the neural net as well.
I don't know that I can say neural nets (or any other machine learning) would do a good job of categorising though.
