Luck is a four letter word.
It is, admittedly, an odd sort of word to pick for St. Patrick's Day. St. Patrick himself has nothing about "luck" attributed to him. There are stories, mostly apocryphal, about his use of the clover as part of his teachings - but this would have been the three-leaf clover, not the four-leaf clover people associate with luck.
The phrase "the luck of the Irish" was coined centuries later, and is either an ironic statement about how UNlucky the Irish have been for most of their history, or specifically relates to the luck of a few of Irish heritage during the American Gold Rush.