One such Jameson family, dating from the 15th century, claims their origins and lineage directly to King James I, of Scotland.

As Ian Jameson, a descendant of this family says: I am a direct descendant of King James 1st and his mistress Janet Gunn, who had two illegitimate sons who held the name "Jamesone." There were once many ancestors who had the name Jameson-Gunn, but I have not seen any recent descendants that bear that name now. Our branch of the family is a sept of Clan Gunn and also a sept of the Butes. We wear the Gunn Clan tartan, as well as the Royal Stewart.

They were ship owners and mariners at Aberdeen. King James gave them the task to arm their ships for an expedition against Norwegians, Pirates and others of the King's enemies, which at that time had infested the North Coast of Scotland. They were both successful in their expedition and did good service for the country. In acknowledgement of this, King James was pleased to grant them an Armorial Bearings as a reward.

For a shield: Three anchors divided by a Fesse wavy.
For a Crest: A Roman Galley armed with sails spread with flags and displayed proper.
For a Motto: "Sine Metu Ad Littora Tendit" = Without Fear He Defends the Coast.

The two sons subsequently split up and the eldest son, of whom I am descended, took on the first part of the family motto Sine Metu, the younger one adopted the second part, Ad Littora Tendit. Basically the entire motto Sine Metu Ad Littora Tendit means Without Fear He Defends the Coast. The Plantagent Lion in the sail of the Roman Galley signified the Royal descent, but illegitimate. This is displayed in the present Coat of Arms that I had matriculated in my name, in Dublin in 1976.

There is no known actual documented proof of this story. However, the story itself illustrates how the surname Jameson can be created in unexpected ways. Although the specifics and details of this lineage are not clear, it can be assumed any illegitimate children of such a union would be known as Jamesons or Jamesones or Jamesone-Gunns, rather than Stuarts or Stewarts, for the obvious sensitive reasons. By this example, it can also be clearly seen how this Jameson family, like many others, may have no relationship with any other family with the exact same surname. 

Furthermore and to add to the irony, the creation of this particular use of the surname Jameson was done within a union of two people whose separate families are closely associated with the only two separate Scottish Clans (Stuart and Gunn) that are known to have Jameson Septs and how this new use of the same surname Jameson was created without connection to either of those existing and established septs. As an aside, it may also be interesting to consider how or with which clan any descendant of this union should claim allegiance.