Was it similar to how surnames based on professions became popular? I'm especially curious about the name Cashdollar because it sounds like it was invented by a hip-hop artist in the '90's.
There are myths about ancestors with particularly strong arms, et cetera, but it seems more likely that it's somewhere between a brag and an adopted group name. In the Armstrong case, it was a clan name for a particular group of border ruffians in contested areas between Scotland and England who repelled invaders and stole (reived/reaved) from others to stay alive, similar to how people engaged in conflict today might come up with impressive-sounding names for their groups. Armstrong was first recorded about 1223.
http://www.armstrongclan.info/clan-history.html
It's also important to remember that various cultures throughout time have different naming conventions. The idea that people have a given name and a family name which passes, unmodified, patrilinealy to children, is not universal. Scottish clans would create and take up names according to allegiance, not necessarily birth, and even have to take new names if their clan were outlawed. Surnames as we think of them weren't common in Britain until between the sixteenth and eighteenth centuries.
My research stops at the history of Armstrong, so I won't speculate on names like Cashdollar.
Reaney and Wilson, A Dictionary of English Surnames, admit that their derivations of meanings of surnames could be speculation. Nicknames were not at all uncommon as a source of surnames in England.
But not all descriptive surnames were complimentary! Vidler 'wolf-face', Chaffin 'bald', Grelly 'pock-marked', Daft 'foolish', Musard 'stupid', Gulliver 'glutton', Greed, Godoenough, Careless, Pennyfather 'miser', Gaine 'trickery', Milsopp. Or a case recently found, the earliest citation of the F-bomb ( http://www.medievalists.net/2015/09/the-earliest-use-of-the-f-word-discovered/ et multa caetera):
Dr Paul Booth of Keele University spotted the name in ‘Roger Fuckebythenavele’ in the Chester county court plea rolls beginning on December 8, 1310. The man was being named three times part of a process to be outlawed, with the final mention coming on September 28, 1311.
Dr Booth believes that “this surname is presumably a nickname. I suggest it could either mean an actual attempt at copulation by an inexperienced youth, later reported by a rejected girlfriend, or an equivalent of the word ‘dimwit’ i.e. a man who might think that that was the correct way to go about it.”
Do beware of misleading etymologies. R&W derive modern Cash from Old French casse 'box, chest to keep wares in' and suggest 'Metonymic for a maker of these'. Cashen from Gaelic Caisin, a diminutive of 'cas' 'crooked'. Or, from the above article,
For example, the name John le Fucker appears in 1278, but this likely could be just a different spelling for the word ‘fulcher’ which means soldier.