The tune itself appears to be published for the first time in Dr Park's Songs, 1876, as MI NIAN DHU.
It is set down by Dr John Park (1804-65) for voice and piano.
Words by John Park.
A note below reads: The Gaelic Air to which the following words are set was given to me by the Rev: John Thompson of Duddingstone, and bears the name of "Mi Nian Dhu" ie My dark Girl.

The tune was arranged by Malcolm Lawson for four-part chorus, in volume I of Songs of the North (Field & Tuer, 1885), with the words by the late Rev Dr John Park of St Andrews.

Prcy Grainger was lent the volume in c1898-1900 and visiting Scotland in June 1900, arranged this song for SAATTBB chorus a cappella (or with piano as solo or as accompaniment) without putting in the words.
The manuscript is in the Grainger Museum Melbourne.

Over fifty years later in 1954 he arranged a shorter piano version from memory. His piano version was published for the first time in 1983 edited by Ronald Stevenson (Edition Peters) with a facsimile of the 1954 manuscript.

The Grainger choral version, first published 1982, superseded now by Bardic Edition, is now on several CD recordings. Mo Nighean Dubh. There are a few transcription changes from Grainger's manuscript.

There are several tunes titled Mi nian dhu, mo nighean dhu, etc, but they turn out to be entirely different melodies with different words.