It's my name too. (Máiréad Máire Ni Riain)
Maggie, how do you pronounce your full name, phonetically?
Mu-raid Maw-re Nee Reen
Most Irish people pronounce Máire as maw-re, some say "moy-ra"
When you see any name with a prefix O' that is the masculine form, meaning "grandson of"
Girls or unmarried ladies, or ladies like me who prefer to keep their maiden name, the prefix is Ní
Our Family name was originally O'Maoilriain which is pronounced O'Mulreen.
It was anglicized to RYAN.
If you know anybody with that surname, it's a safe bet they had Tipperary ancestors.
Go there and look at the names over the Pubs, you'll see what I mean!