For some, a paddy is where rice is grown.
via merriam-webster: St. Patrick's Day, while originally a feast day for the patron saint who brought Christianity to Ireland, has evolved into a secular celebration of Irish heritage. As its reach has widened, many are familiar with the nickname "St. Paddy's Day," where "Paddy" is the shortened form of Patrick, which itself comes from Pádraig. While the holiday is sometimes spelled "St. Patty's Day," some find this to be offensive as "Patty" is a shortened form of the female name "Patricia."
Patrick is the Anglicized form of Pádraig, nicknamed Páidín and Paddy.