Fowler reckon's that Minch is probably from th Old Scandinavian "megin nes" = great headland.

They give "mickill ey" (muckle island) for Mingulay (Miughalaigh) so the 'n' looks as if it intruded on Anglicisation.

Of the song they say "A famous but inauthentic 'folk song', based on a tune from the Western Highlands and with words by Hugh S. Robertson, written in 1938 for his Glasgow Orpheus Choir."

As for non-Scandinavian roots, you wouldn't find many Anglo-Saxons in the Highlands and Islands till fairly recently, and their place-names are usually recognisably English rather than Anglo-Saxon.