If by this you mean you're interested in the responses of the rulers of Scotland and the Welsh principalities to the Norman Conquest then the answer is that, yes, there were attempts to take advantage of the new political situation. It isn't really wise to see this in terms of "re-conquest" - the situation was far more fluid than this, some of the actions that were taken were responses to moves made by the new Norman rulers of England, and, in any case, and raids and the threat of raids were at least as useful as political tools than outright attempts at long term occupation of territory. Similarly, while English kings had no formal claim to Scotland or the Welsh principalities, they claimed overlordship and frequently interfered in Scottish and Welsh affairs
Anyway, a brief summary of events requires us to go back to the Anglo-Saxon period to trace the antecedents of the trouble that flared after the Conquest:
We know that King Cnut took the submission of Malcolm II in 1031, and there is also some evidence that Malcolm's grandson Malcolm III Canmore spent some time at the English court during the reign of the usurper Mac Bethad (Macbeth), who ruled Scotland 1040-1057. There's a good blog piece by a scholar unravelling this aspect of Scottish history here:
We also know that Norman knights first appeared in Scotland during Macbeth's reign – as early as 1052, in fact, when they were expelled from England by Godwin’s victory over Edward the Confessor. Two years after that, Siward, the Earl of Northumbria, led an English invasion resulting in a major battle at which most of Macbeth's mercenaries seem to have been killed - one of Siward’s sons also died. This in turn made it possible for Malcolm III to invade. Macbeth was killed at the Battle of Lumphanan and after a brief period during which his inept son Lulach (sometimes known as Lulach the Fatuous) held onto the throne, Malcolm became king – owing, therefore, a debt to the English for his throne. Although there are few traces of English influence at his court, he did give sanctuary to Harold's brother Tostig in 1066.
It took Malcolm another two years to make a move that might be interpreted as a response to Hastings: in 1068 he secured his northern frontier by marrying the daughter of Earl Thorfinn of Orkney, and in the same year he gave sanctuary to the Anglo-Saxon pretender to the English throne, Edgar Atheling and to Edgar's mother and sister. They used Scotland as a base to lead a revolt in the north in 1069.
In 1070, with his Orkney wife dead, Malcolm married Margaret, sister of Edgar Atheling – the future St Margaret. This does seem to have resulted in the importation of significant English influence to the Scottish court. Scottish royal names became increasingly Anglicised (Alexander, Edgar, David); Anglo Saxon culture predominated; and so far as we can tell, the relatively small number of Anglo-Saxons at Malcolm's court enjoyed a disproportionate influence.
It is very hard, in the absence of detailed sources, to know what Malcolm wanted to do on his southern border. It's been suggested that his aim was to re-found the old Kingdom of Bernicia (between the rivers Tweed and Tay). In any event, the situation in the north changed after 1072, when King William’s Harrying of the North created a political vacuum by seriously weakening the power of the Earl of Northumbria and the Archbishop of York. This encouraged an invasion by Malcolm, who devastated Durham and Cleveland. William responded by assembling an army and marching north in the summer of 1072. He reached Perth, in the heart of Malcolm’s kingdom, while his fleet sailed up the Tay estuary to meet the army (this rare example of a Norman "combined operation" was a large risk and hence suggests how serious the Scottish situation seemed to be.)
This invasion succeeded in bringing Malcolm to terms. He met William at Abernethy, recognised him as King of England and swore fealty. Malcolm also expelled the dangerous Edgar Atheling from his court; Edgar was forced to leave the country, though he later did return to Scotland.
A seven year truce was signed and it ran until 1079, when the Scots promptly launched a large scale raid into Northumbria.
In 1080 William sent his eldest son, Robert Curthose, to subdue the Scots. He held a council at Falkirk which reiterated the terms of the Treaty of Abernethy; it's possible to argue that Curthose's unwillingness to fight was tacit recognition of Scottish strength. In any case, Falkirk marks the high watermark of Malcolm’s power. The Normans took steps to make raiding harder and less profitable for them thereafter; as he retreated, Robert built a fortification at a place he called New Castle.
None of this stopped the Scots from attempting to profit from any sign of weakness to the south. Most spectactularly, when the Conqueror died in 1087, Malcolm attempted to take advantage of the dispte over the succession by refusing to recognise William Rufus as legitimate king of England. Instead he did homage to Curthose, and went on to invade Northumbria for a fourth time in 1091, forcing Rufus to raise an army and march it to the Tweed. This brought Malcolm to terms and he acknowledged Rufus as king and swore fealty. The pesky Edgar Atheling was expelled from Scotland again; he went south with Rufus, who in 1092 returned to the north, built a castle at Carlisle and peopled it with incomers from the south and Durham. This was in an area that had long been Scottish territory, or at least claimed by Scotland and it moved the English frontier north. With the construction of Newcastle on the east side of the country it also materially improved England’s defences.
Malcolm promptly claimed that Rufus had broken their treaty. He travelled to Gloucester for negotiations – where Rufus slighted him, perhaps intentionally provocatively. Malcolm returned to Scotland and attempted another border raid but he was killed in an ambush at Alnwick together with his son and heir, Edward.
There followed a confused interregnum involving Donald II (Shakespeare's Donalbane) and it was not until 1097 that Malcolm’s younger son Edgar was confirmed as king. Scotland would not be a significant threat again until the reign of David I more than 30 years later.
Confused? All in all that's the summary of a non-untypical 40-odd year period of Anglo-Scottish relations in this period. The tl;dr is that it's a record of short-term policy and temporary shifts in the balance of power – certainly nothing so grand as a concerted attempt to conquer and hold on to territory.