Whilst I can't comment on how likely it is for your family to have Norman roots, I can tell you that plenty of Normans settled in Scotland. Norman knights played a significant role in medieval Scottish history, both militarily and politically. The famous MacBeth for instance used Norman cavalry in the civil war, whilst his opponent, Malcolm III (1058-1093), and his sons either recognised the superior power of Norman England or were placed on the throne by their Norman backers. Two of Malcolm's sons (both kings), Duncan and David, were knighted by Robert Curthose (son of William the Conqueror) and Henry I respectively, whilst their sister Matilda married Henry I. The government of these kings therefore was increasingly Anglicised - they brought in Norman settlers, they followed English customs and sought to reform their administration along English lines. David I is an especially good example, as he was partially raised in England and granted significant lands there (most importantly the Earldom of Huntingdon). He intervened in the English Anarchy for both personal gains and to support the Empress Matilda, even taking over northern England for a bit - because of how Anglo-Norman his administration was, arguably he just stepped into the void created by the Anarchy and governed northern England as a Norman king would!

Other settlers, most prominently the Flemish, were also introduced, but the Normans stood out in power and prestige in the sources, possibly because many of these families owned land in England as well and so had a great deal of wealth/influence to use in Scotland. In the 12th Century, English chroniclers were quite happy to describe Scottish knights/kings as noble and chivalrous even when writing about Anglo-Scottish wars, but at the same time they described many Scottish soldiers, particularly those from less Anglicised regions such as Galloway, as barbarians. Later, the families that dominated Scottish politics in the Wars of Independence, the Balliols, the Bruces and the Comyns, were all descended from Norman families granted land by earlier kings, but by then their outlook was rather different. Unlike during the Anarchy, when Robert Bruce the elder fought on the side of the English and Robert Bruce the younger fought on the side of the Scots, there was no longer a conflict of loyalty - the men fighting the Wars of the Independence were thoroughly Scottish, as they and their families have been immersed in Scottish politics for centuries.