BAD NAME. As cute and nice as it sounds, it's considered to be an perjorative term for an Irish person in British and American slang (the word "paddywagon" is regarded to be dervied from this nickname, from the Irish being arrested so often in big cities (many times for racist/prejudice reasons)).
I agree with the previous poster. Although, Paddy is a sweet name, it is very offensive. However, I believe the term paddywagon came into use because many Irish immigrants became police officers.
There was a restaurant in Cork City, Ireland named "Paddy Garibaldi's". (There was or is another in Kinsale also and perhaps elsewhere. I don't know which was or is the original.)
Paddy Boom, born Patrick Seacour is the crazy drummer of the band Scissor Sisters. Although his outfits would argue, he is the only straight man of the group. Paddy described joining the band as "jumping on a fast moving train". Paddy recently lost his mother to Cancer, so took a year out of Scissor Sisters.
Previous users have pointed out the offensiveness of this name, but as an Irish person who's lived in Ireland my whole life, I don't think that really matters too much here as it is so common but maybe in other countries it does. I know a woman called Paddy too, I always found it a bit strange though.