According to Grammar Girl:

Each year, millions of Americans annoy the Irish by referring to St. Patrick’s Day as St. Patty’s Day. For the record, the correct short form is St. Paddy’s Day. According to the Provisional Government of Paddy, not Patty, the d‘s in Paddy come from the Irish spelling, Pádraig . Visit their site to be chastised in a proper Irish manner.