According to the History Of Clan Stevenson at the website Electric Scotland, "The name is first recorded in Scotland in 1388 and means 'son of Steven'. The family originally farmed land in the parish of Neilston, Renfrewshire but they spawned a host of famous figures in history: Robert Stevenson (1772-1850) and his son Alan (1807-1865), David (1815-86), and Thomas (1818-87), father of Robert Louis Stevenson (1850-94), novelist, short story writer, and poet."
Neilston, Renfrewshire is indeed the home of the earliest Stevensons identified in our line.The following tradition of the origins of the STEVENSON name originated with Robert Louis Stevenson: The MacGregors of Scotland were proscribed, or outlawed, and so took other names to avoid persecution by the Campbells. Stevenson meant "Son of My Love", that name being given to one of the MacGregors who was born in hiding on a hillside in Scotland. If this legend is true, since our Stevensons are from the same small village as the earliest ancestors of Robert Louis Stevenson, our line may also descend from the MacGregors. Other explanations of the origin of the Stevenson name follow this version. Here is the story in the words of Robert Louis Stevenson:
This Family "according to tradition is one of the proscribed of the clan MacGregor of Glenorchy, who was born among the willows in a hill-side sheep-pen - 'Son of my Love'. The heir of MacGregor of Glenorchy, murdered about 1353 by the Argyll Campbells, appears to have been the original 'Son of my Love' ; and his more loyal clanmen took the name to fight under. It may be supposed the story of their resistance became popular, and the name in some sort identifies with the idea of opposition to the Campbells.
"Twice afterwards, on some renewed aggression, in 1502 and 1552, we find the clan MacGregor of Glenorchy again banding themselves into a Sept of the 'Son of my Love', and when the great disaster fell on them in 1603 the whole original legend re-appears, and we have the heir of Alister of Glenstrae born `among the willow' of a fugitive mother, and the more loyal clanmen again rallying under the name Stevenson.
"A story would not be told so often unless it had some basis in fact ; nor - if there were no bond at all between the "Red" Macgregors and the Stevensons - would that estraneous and somewhat uncouth name be so much repeated in the Legends of the 'Children of the Mist' (the MacGregors).
"Doubtless a proscribed clan could not be particular; they took a name as a man takes an umbrella against a shower; as Robert (Rob Roy) MACGREGOR 1671 - 1734 took Campbell, and his son took Drummond. But this case is different; as Stevenson was not taken and left - it was consistently adhered to. It does not in the least follow that all Stevensons are of the clan Alpin; but it does follow that some may be."
Robert Louis Stevenson thus comes to this conclusion : "I cannot conceal from myself the possibility that Nathanial Stevenson 1732 - 1824, my first authentic ancestor, may (therefore) have had a Highland alias upon his conscience and a claymore in his back garden."
The Founding Ancestor of Robert Louis Stevenson's Family is Nathanial Stevenson 1732 - 1824, of Craig, Parish of Neilston, Co. Renfrew, Scotland, Tenant Farmer in Carsewell (1653). While a definite link to our Stevenson family has not yet been found, both families lived in the small locale of Neilston Parish, County Renfrew, Scotland at the same time and shared comm on family given names; this would indicate a reasonable degree of probablilty that the two families are related.
Adding at least some credence to Robert Louis Stevenson's story is the following, from Stevenson researcher Bill Stevenson: "Following the proscription of the MacGregors, many MacGregors found it necessary to change their name, e.g. to Gregor, Drummond, Grant, Campbell, Stevenson. These names were often those of relations, friends, neighbours, landlords, etc. Scottish history books, articles and theses are full of such occurrences."
However, Scottish Genealogist Gordon Johnson offered this opinion: "The supposed origins of the surname in Clan MacGregor seems somewhat fanciful, as the surname is found in all parts of Scotland by the 1400s. As it means simply "son of Stephen", just as Johnson is "son of John", the surname will have arisen many times in different localities and it is highly unlikely that any family line can be traced back to one specific origin. Stephen itself is a Biblical name and was introduced to Scotland by the Normans. The Highland Stevensons would probably have no connection to those in the Lowland areas of Scotland."

The following is from ""The Stevenson Family, a Record of the Descendants of James Stevenson, Burgess of Paisley in 1753", by Hew Shannon Stevenson:
" 'From the thirteenth century onwards,' wrote Robert Louis Stevenson in his Records of a Family of Engineers, 'the name, under the various disguises of Stevenstoun, Stevensoun, Stevensonne, Stensone, and Stewinsoune, spread across Scotland from the mouth of the Firth of Forth to the mouth of the Firth of Clyde.' Although apparently the distribution of the name seems to have been rather more widespread than this would indicate, most Scottish Stevensons have their origin in the counties of Lanark, Renfrew and Ayr.
The first recorded instance of the name in this area occurs in the Ragman Roll of 1296 when Johann de Steueneston swore fealty to King Edward the First after the wars of those times [note: this would be between about 1290-1296]. His descendants continued to occupy the house and lands of Stevenston (or Stevenson as it came to be called) and to play their part in the pattern of history with more or less distinction, serving as jurors, commissioners for war and witnesses to deeds, until the last of the direct line, John Stevenson of that Ilk, died in 1670 after which the estate was sequestrated "for the behove of the creditors". [A footnote on RLS reads: "Robert Louis Stevenson, although belonging to a different Stevenson family, was related to all the descendants of Nathaniel Stevenson (1732-1824) through the latter's mother-in-law." Under the entry for Nathaniel Stevenson: "Nathaniel Stevenson m. Janet Maxwell (1728-1821) in 1766, daughter of Robert Maxwell of Westbrae, Paisley, a descendant of the Maxwells of Bredieland and Castlehead. Her mother was Janet Stevenson, of the same family as the author Robert Louis Stevenson".]
The estate of Stevenson, "albeit not great", is situated in the barony of Bothwell, near Motherwell in Lanarkshire. It would seem more likely than not, though nothing has yet been proved, that our own James Stevenson, Burgess of Paisley, was descended from a cadet branch of this family. Dalziel, the alleged place of his origin, lies only three miles distant from Bothwell, and in an age when it was customary for younger sons to migrate to nearby parishes, there being no room for them in the paternal house after they had grown up, such a speculation would not seem unlikely.
Of James Stevenson's immediate ancestry nothing is certain. The registers in the parish of Dalziel, and the neighboring parishes of Cambusnethan, Shotts, Hamilton, Bothwell and Avondale, where they go back far enough, give no trace whatever of his birth or parentage. The first authentic record does not appear until 1732 when the baptism of his son Nathaniel was recorded in the Avondale Parish Register.
The information, such as it is, about James Stevenson has come to us from the personal recollection of one of his grand-daughters, Jane Stevenson (1767-1862) who, when a very old lady, told all she knew to one of her nieces who wrote it down. She was certain that her great-grandfather had been a farmer, and nearly certain that he had lived in Dalziel. But she could not recall his name. She thought it had been either Nathaniel or James, but had forgotten which.
Although the Burgess of Paisley's ancestry must remain unknown, his descendants certainly have been prolific. His progeny number no fewer than 525, of whom some 350 are living today."

And finally, this version from Stevenson researcher, Geoffrey Stevenson: "Stevenson appeared in the 11th Century in the Edinburgh area. Supposedly, a daughter of William the Conquerer was married off to a Scottish Prince. Reputedly she had spent time at the Court of Hungary where the Patron Saint was St Stephan, and she named one of her children after the Saint. As often happened, peasants thereafter named children after popular royal persons, and Stephan and it's variants got started. I also saw a monument to two Scottish princes who were at the Hungarian Court at Estahazy about that time and it would tend to confirm the above story. "