According to my dictionary, the term comes from the refrain "Will you come up from Limerick?", which was sung between nonsense verses at parties in the 19th Century.
Another theory is that the comic verses were sung by Irish mercenaries, known as the Wild Geese, returning to their native Limerick after fighting in Europe with the great Catholic soldier Patrick Sarsfield. They certainly became famous - often infamous - as drinking songs in Irish pubs and the form spread to England. There, they came into fashion after being taken up by Edward Lear in his Book of Nonsense.