One of these names, Pollok, is my grandmother's name.  There are various spellings of Pollock, and various stories about how the name originated.  
The family Pollok is said to have taken the name in the 12th century when they were given lands in Scotland by King David I of Scotland, along with other
Anglo-Breton refugees from a succession war in England.  Clan Pollock website says that they took the name after the name of the location where they
settled.  I think a possible explanation is that they took this name from a combination of the French which they spoke and the local name for lakes.  Thus:
"peu" (small) de Lachs (lakes in Gaelic).  

And then there is the frequent claim that the modern Scottish name Pollock and the Jewish name Pollak have different roots.  

I put this snippet of information forward to  show that names are deceptive, and one cannot be entirely sure where they came from.

But other details of the history of the Polloks makes me think they easily could have been of partial Jewish descent.  Was this descent maternal or paternal or both?
In any case, very mixed.  They were not wholly Jewish even if they were partly Jewish.  Which "counts" in determining descent from David, the maternal or the paternal line?  In Bible times, it would have been the paternal line, but would it be the maternal line today, which is the usual descent counter among modern rabbis?

The founder of the Pollok line in Scotland, Peter, sometimes signed his name on documents as "Peres", which is not only a Jewish name from the Old Testament,
but it the name of one of Judah's twin sons by Tamar.  I have found documents showing that this name "Peres" or "Perez" was repeated in generations of the family living in Renfrewshire and Lanarkshire.  Usually the people in question were "sheriffs", so there were documents preserved about their activities, and in the early,
wealthier years they gave land to various priories and abbeys in the area, and the donation records listed their names.  I recently saw a study online by a Scottish
Ph.D. discussing the INTENSELY VARIED racial and cultural backgrounds of the people who lived in early medieval Scotland, particularly in the midland and borders regions.  Bingo!  Corroborates what I've seen elsewhere and put together myself as a thesis.  There is lots more where that came from.

Well, Doves, this may be of little interest to you considering the momentous times and struggles we face, to discern who has a tiny bit of King David inhabiting his or her genes.  I just wanted to say that this IS interesting, very interesting to me when I try to figure out why I am the way I am, but it is as nothing in our consideration of our salvation.  Being a child of God is THE important thing, not whether any of us are descended from David or from other Jews of his period.  

But I am grateful for the list of modern Jewish people's names who are descended from David.  Thanks!

Mariel  (I just found out from the Scottish Ph.D. that my "aunt" in the early middle ages actually spelled her name Muriel rather than Mauricle, as most weblinks have it.  Pretty close to my Mariel.  The proof is the documents collected by the Ph.D. referencing donations by the Pollok, with Muriel Pollok being a donor, as she
was the heir of Peter (Peres) de Pollok.