This is the entry for Herbertson in The surnames of Scotland, their origin meaning and history, by George Fraser Black, New York : New York Public Library & Readex Books, 1946.
HERBERTSON, ‘son of Herbert,’ from OE. personal name Herebeorht, meaning army bright. Archibald Herbertson was burgess of Glasgow in 1525 (1), and Robert Herbertsoune was notary public there in 1536. The name is common in the Glasgow Protocol books of the sixteenth century as Herblsone (1551), Herbisoune, (1550). George Harbertsone, is recorded as a witness in Glasgow in 1559 (2) William Herbesone of Nortb Berwick is mentioned in 1555 (3), and Richard Harbertson, son of Jobn Harbertsoun, notary, was burgess of Glasgow in 1605 (4). Margaret Herbertson is recorded in the parish of Monkland in 1619 (5), and Martin Herbertson was tenant in the barony of Mousewall (Mouswald, Dumfrlesshire) in 1673 (6). An action at law against Georg Harbiesone in Hairshaw, 1706 (Corsehill, p. 213). Harbersone, Harbertsone, Harbeson. and Harblsone 1706, Herbertsoun 1609. (1) Registrum episcopatus Glasguensis... Edinburgh, 1843. 2 v. p. 497.Commentary: it is obvious the spelling of the surname that was originally Herbertson was highly variable even before it ever left Scotland. The t, which is often glottalized (even by myself) in North-of-England and Scottish dialects, therefore is prone to disappear in the mouth of someone who doesn't use glottal stops. Thus do the Haole mispronounce Hawai'i. And while Scots do not tend to lose the 'r' (in fact, they rrroll it) the English do.
(2) Highland papers. Edited by J. R. N. Macphail, Edinburgh, 1914-34. 4 v. p 203.
(3) Carta Monialium de Northberwick... Edinburgh, 1847, p. 70.
(4) Inqvisitionvm ad capellam domini regis retornatarvm, qvae in pvblicis archivis Scotiae adhvc servantvr, abbreviatio. 1811-16. 3 v., p. 75.
(5)The commissariot record of Hamilton and Campsie. Register of testaments, 1564-1800. Edinburgh, 1898.
(6) Proceedings of the Society of Antiquaries of Scotland. Edinburgh, 1851-1938.
(7) Corsehill baron-court book, 1666-1719. (In: Ayrshire and Galloway Archaeological Association. Collections, v. 4, p. 213).
As for the name itself; it might have arrived in the Scottish lowlands via the major settlement of southern Scotland in the 11th and 12th centuries by the English, which Black (loc. cit) attributes to the Norman push north into Lancashire and Yorkshire. He says there is no evidence Gaelic was spoken in southern Scotland outside of Galloway after the mid-12th century. There are however, no Harbisons, or Herberts, or any variation thereof, in the Ragman Rolls of 1291-1296.
Black suggests the use of -son suffix as a patronymic began in the early 14th century.