Hmm, I think that the story of Clan Morrision comes from at least 3 different strands. First of all there are what is probably the most common branch, those who are descendents of a man or men whose name was Maurice, which was hugely popular in Medieval times. This branch is based in Aberdeenshire, (I am from there and have several Morison ancestors. The name was written with one R until very recent times, although some still use it that way).

The West Coast/Hebridean branch settled in Skye and Mull as bards under the patronage of the chiefs of Clan MacLeod (around 1600)and MacLean of Duart (around 1612) respectively. The original Gaelic spelling for this family was Ó Muirgheasáin, as they were from the North of Ireland where they had previously held bardic positions for the O'Neills.

The Isle of Lewis branch claimed descent from the King of Norway, and in Scottish Gaelic the spelling was Mac Gille Mhoire, which is probably closer to what you refer to. This clan seemed to disperse after 1613, following a huge defeat by Clan MacAuley after years of inter clan warfare, and it was declared in the 19th century that it was impossible to trace their descendants.

Although apparently not related, these 3 families were declared a united clan under one chief by the Lord Lyon in 1965. So,
Mac Gille Mhoire was the original Gaelic name of one branch only of the Morrisons, the branch where, so far, no descendants have been traced.