actually, the name is spelled correctly in the lyrics in the first message, albeit not in the thread title.
according to MacLennan's Gaelic dictionary, "áirigh" is a shieling or sheep pasture and "airidh" means worthy, meretorious. Fionn is fair-coloured. There is also a word "fionnairidh" in the dictionary meaning either watching or evening (I think because another meaning of fionn is cool and aire can mean to keep watch).