There are more or less two groups of Livingstons in the world today. One is the highland Clan Livingstone, previously Clan MacLea. The men in this group are not a single family, but many families brought together by socio-political allegiances under the leadership of a chief. The other Livingston group is the once wealthy and powerful lowland family who lost their lands and titles in the 18th century. A branch of the family became prominent in the colonial United States, and the list of influential lowland Livingston descendants includes Walter Scott, Lewis Carroll, Winston Churchill, Franklin D Roosevelt, the Bush presidents, and many others.

Why the shared surname? No one knows. All that is known for sure is that, sometime in the mid 1600s, the MacLeas formed some kind of alliance with the lowland Livingstons that allowed the former is use the name of the latter. Wikipedia:

In the mid seventeenth century James Livingston of Skirling, who was of a branch of these Lowland Livingstons was granted a nineteen-year lease of the Bishoprics of Argyll and the Isles. Sometime before 1648, James Livingston seems to have stayed at Achanduin Castle on Lismore, and it is thought that around this time that the surname Livingstone would have been adopted by MacLeas on the island.

The only evidence for any genetic connection between the highland the lowland Livingstons is a 1742 document, penned by a Mr Duncan MacLea, Minister at Dull. Duncan MacLea writes that a Malcolm MacLea told him that a Lord Livingston told him that, despite the disparity in their socio-economic status, the Livingstons were actually descended from the MacLeas in the highlands. This document doesn't satisfy any of the criteria for a historically reliable source: its author is a MacLea who is talking up his clan; there are no other documents that support the author's claim that the lowland Livingstons are former MacLeas; and in particular there are no writings from any lowland Livingston that support this claim. Indeed, lowland Livingston-historian, and descendant of the lowland Livingstons, E.B. Livingston, wrote of the eponymous founder of the lowland Livingstons, Leving of Leving's Town, that he was "doubtless" of Saxon lineage.

But in any case, there are genetic and historical reasons to think both theories are wrong. First genetics. There is a family group of Livingstons, whose ancestors lived in the lowlands, and whose members carry Y-DNA that belongs to the Eastern European haplogroup R1a. While rare in Scotland generally, this haplogroup is relatively common amongst Scottish aristocrats. The most well-known carriers are the MacDonald chiefs, whose origin is thought to be Norse rather than Gaelic, and Big-Y results for the R1a Livingston men tested so far appear to confirm Viking ancestry (the top match recorded by Y-Full is to a Viking skeleton unearthed in Orkney (VK205)). History is in accord with genetics here. Many lowland Scottish families are of Norman origin; the Normans were former Vikings; and there is evidence for a Norman origin of the lowland Livingston family. It is difficult, in the light of the following facts, to seriously imagine that the eponymous founder of the lowland Livingstons was not a Norman:

• the name "Leving" is a variant of a name that appear on the Battle Abbey Rolls (Loveyne, Lovein, Lovan, Leuuin, Lieuvin), i.e., Leving is a potentially Norman name

• Lieuvin was a district in Normandy at the time of the invasion (and remains so to this day)

• Leving or his forbear appeared in Scotland around the time when land was being confiscated from Anglo-Saxon aristocrats and given to the Normans (late 11th century), and there is no evidence of any "Livingstons" in Britain prior to the Norman Invasion

• Leving gave his sons Norman names (Thurstan, Hugh, and German or Germanus)