At the Battle of Dunnichen or Nechtansmere in eastern Scotland on 20 May 685, the Pictish army under their king Bridei III overwhelmingly defeated the Northumbrian army of King Ecgfrith.
http://en.wikipedia.org/wiki/Battle_of_Dunnichen .
An alternative scenario is that the Northumbrians decisively win that battle, with King Bridei being killed, and conquer the whole of OTL Scotland (perhaps excluding the Northern and Western Isles: The Orkneys Isles, the Shetlands and the Outer Hebrides) by the year 700.
This kingdom of Northumbria successfully resists the power of Mercia in the 8th century and attempted Danish/Norse/Wessex conquest in the 9th and 10th centuries. Though it might lose some peripheral territory to the Danes/Norse.
By the 11th century the kingdom of Northumbria with its capital in Edinburgh includes all or most of OTL Scotland and extends as far south as the Humber and the Mersey. It is prosperous, and militarily an equal match for the kingdom or kingdoms to its south.
In the following centuries it is able to resist attempts at conquest from the south. It conquers or reconquers the Northern and Western Isles from the Norse. The Gaelic speaking clans of the Highlands and Islands are subdued and brought under the control of the government in Edinburgh.
There would most likely be marriages between the royal houses of Northumbria and England (assuming that a kingdom of England develops), though not necessarily a union of the crowns as there was between England and Scotland in OTL.
This Northumbria would be wealthier than OTL Scotland so it would want to maintain its independence from England. It would include the coalfields and manufacturing industries of OTL northern England. It would probably want to establish an empire outside Europe.
In the present day it is still independent, though a member of the European Union (or similar organization).