Surely the songs to sing on St Patrick's Day would be songs about Irish saints?
In Ireland we usually sing the hymn "Hail Glorious Saint Patrick", which starts:
Hail Glorious Saint Patrick, dear saint of our isle On us thy dear children, send down a sweet smile..."
to which the saint invariably replies not by hailing but by raining heavily on the parade.
Or what about "In Glendalough lived a young saint, in odour of sanctity dwelling, an old-fashioned odour which now, you seldom or never are smelling".
You'll find the first in any collection of Irish hymns, the second was recorded by The Dubliners.