Well, those two names are new to me, as far as their etymology is concerned. Still, the mechanism is very reminiscent of the etymology of orange (by all accounts, from 'a norange', newt (again, by all accounts from 'an eft' via 'an ewt' or maybe via 'a neft') and lots more exx. with an+V- ---> a+ nV-, or a + nV- ---> an + V-.
I had not seen examples of a possible generalization of the 'floating n' phenomenon fairly common in the indefinite article with vowel- or n-initial words before, if indeed that is what Nash and Rose are.
James L. Fidelholtz
Graduate Program in Language Sciences
Instituto de Ciencias Sociales y Humanidades
Benem'erita Universidad Aut'onoma de Puebla, M'EXICO
When I was taught placename etymology, over 40 years ago, we were told never to guess. This applies to surnames too. The etymology of names can be unusual and creative. So to know the etymology you have to have traced the name back through documents until you get to its source.
Ancestry.com says that the surname 'Rash' comes from Old English (and here AE stands for the letter ash, which looks like a and e squashed together, and TH stands for the letter thorn) "AEt THAEre AEsce": at the ash. The ending -re is an inflection controlled by the pronoun. I assume these are the etymologies of the place names from which the surname comes. They are plausible, but you would need to look at the full evidence to see whether they are right.
Ancestry.com also say the name is 'Jewish'.
Remember that surnames did not emerge in England until after the Old English period, after 'the' had lost its inflections. Owing to massacres, few Jewish people survived from medieval England.
If the name is Jewish, it is more likely to be an English sounding name based on a name of some other origin.
Anthea Fraser Gupta