Mysha, according to Wikipedia : " 'Mingulay' is derived from Mikil-ay, the Old Norse for 'Big Island' "

http://mysongbook.de/msb/songs/m/mingulay.html:
'[1995:] A large number of the place names in the Barra isles are of Norse origin (or are compound Norse-Gaelic), as are the names of the islands themselves. The name Mingulay is thought to derive from the Old Norse 'mikil', meaning big, and 'ay', meaning island. In Gaelic it is 'Miughalaigh', pronounced something like 'me-ul-eye', or 'Miùghalaigh', which accounts for the form 'Mewla' given in a 17th century source. [anon 1620 in J L Campbell, ed, "The Book of Barra", 1936, p 44] Monro's version of 1549 - Megaly - is the earliest known; Martin Martin, 1695, gives 'Micklay'. The current spelling and pronuciation in English has drifted further from the Gaelic than in other cases, possibly because of the various forms used by early writers and map makers. [...] Mingulay's most famous song - outside Barra and Vatersay that is - is "The Mingulay Boat Song'. But neither the words nor the melody originate anywhere near Mingulay; it is a romantic invention of the 20th century. It was devised in 1938 by Glasgow-born Sir Hugh Roberton, who was very fond of the melody of 'Creag Ghuanach', a song from Lochaber, which celebrates a crag near Loch Treig. He needed a sea shanty, and so he adapted the music, chose the romantic name Mingulay, and composed the words. It was to be sung in F, slowly and rhythmically. [Roberton Publications, personal information; Derek Cooper, "The Road to Mingulay: a View of the Western Isles", London, 1985] (Buxton, Mingulay: an Island and Its People, pp 33, 47f., )'