The Origins of the Name and of the Clan
The name MacKenzie - which is first documented in 1463 as MacKennych and is given on the 1587 "Roll of Landlords" (referring to the chief) as Mackanyie - is a phonetic English version of the Gaelic MacCoinnich; i.e. "Son of Coinneach”. Since the Gaelic forename Coinneach (or Cainnigh as it appears in some old genealogies) is generally considered the equivalent of “Kenneth”, the name has occasionally been Englished as Kennethson. See the list of spellings of the surname here.
The earliest surviving account of the MacKenzies is the Genelach Cloinni Cainnig (“Genealogy of Clan MacKenzie”) that forms part of the late 14th century collection of pedigrees now known as MS1467. The names on this genealogy are not always clear - it being written in a medieval script which has faded over time - and four versions of it have been published since it was discovered in the early 19th century. All four versions are given and discussed in an article by the Gaelic scholar Ronald Black which was originally published in West Highland Notes & Queries, Series 4, No. 8 (November 2018), and which can be found on his MS1467 website as a PDF here.
There are two Cainnighs in this pedigree, and there's nothing to indicate from which of them the clan is supposed to have taken its name. Since the second of them would have lived towards the end of the 14th century, it appears more likely the clan takes its name from the first of them, who is thought to have lived earlier in the same century. He is remembered in some early clan histories as Coinneach na Sroine or "Kenneth Nistroin from his great nose" - a feature not uncommon in the clan to this day. The matter is complicated however by the possibility that there was a third Coinneach in the clan's ancestry - if, as the distinguished 2oth century Gaelic scholar and historian Rev. William Matheson suggested, the MacKenzies shared a descent from the progenitor of his clan, Mathghamhain/Mathan, whose son Coinneach is on record in 1262/3. See the Matheson pedigree in MS1467 here.
According to MS1467, Coinneach na Sroine was the 3xgreat-grandson of a man called Gille Eoin na Aird (Gilleoin of the Aird) who it also shows to have been the ancestor of the Mathesons, though the link is further back in the pedigrees than suggested by Willie Matheson. The Aird is a fertile, and therefore valuable area to the west of Inverness, stretching into the south-eastern corner of the county of Ross, where all the main seats of the MacKenzie chiefs were later to be found. Though there is no contemporary record of Gilleoin, he or Gille Eoin Og (Gilleoin the Younger) seem to be referred to in the patronymic of one John of the Aird, who’s documented in the late 13th and early 14th centuries. For The Aird see the map here.
MS1467 also traces a clan called Gillanders (Gille Aindreas - "Devotee of [Saint] Andrew”) back to Gille Eoin ("Devotee of [Saint] John), and this is sometimes said to have been the original kindred name of the old Earls of Ross from whom the first MacKenzies and the Mathesons held their lands in Kintail. The Genelach Cloinni Ainnrias traces Gilleoin of the Aird back to Cormac mac Airbertaich, who MS1467 says was the progenitor of many other Highland clans, and a descendant of the famous Scottish king Macbeth. He belonged to the Cenel Loairn, a branch of the original Kings of the Scots who had moved up from their ancestral lands of Lorn, in Argyll, to become the rulers of Mureb (Moray), a kingdom that at one time stretched from modern Aberdeenshire to Ross with a seat at Inverness. See a tree showing possible connections here.
This traditional account of the MacKenzies' descent from the Lords of the Aird and the Kings of the Cenel Loirn was replaced in the 16th and 17th centuries by the legend of Colin FitzGerald, which married an Anglo-Norman noble from Ireland into the Mathesons to obtain Kintail and the name Coinneach/Kenneth. When it comes to the name Coinneach however, one of the clan's most recent academic historians - Aonghus MacCoinnich of Glasgow University - has pointed out that this is a name almost exclusively associated in early medieval times with Argyll, and has suggested that the MacKenzies may have come from that part of the Highlands rather than from Ross [see Ronald Black's article on the MacKenzie pedigree in MS1467 which is referenced above].