This 'public service announcement' exists:
Why on earth would it be Patty rather than Paddy, I hear you ask? Well, it's a simple case of homophony. Two identical-sounding words, one pronunciation, confusion abounds.
What's that? They're not homophonous? Not to you, maybe. Yer average British English speaker will pronounce these with a /t/ and a /d/ respectively, or possibly a glottal stop instead of the /t/. Yer average American, on the other hand, will have what's called a 'flap' or 'tap' instead of the /t/, and this sounds so much like a /d/ it's pretty much indistinguishable and they're pronounced exactly the same.
I'm ashamed to say I don't know how yer average Irish person would say them - my instinct is telling me there's variation between the flap and the /t/ and /d/ distinction, but I don't know. But then any Irish person is likely to know how to spell the name of their patron saint in any case, one would hope.