What variant spellings have you found of the surname in your records? Those might possibly provide a clue. I myself was able to find Ahrenspries in this document from the government of the Falkland Islands:
I wonder if it could perhaps be a corruption of the German word Ehrenspreis meaning "prize of honour". Alternatively, it could come from Ehrenpreis, the German name for the Veronica plant. See its German article on Wikipedia:
The change from E- to A- might have happened through the latter Ä, which is pronounced the same as E and so the two letters could be used interchangeably when trying to transcribe the same sound. As such, the process might have been like this:
Ehrenpreis --> Ährenpreis --> Aehrenpreis --> Ahrenpreis (once your ancestor or one of his own ancestors went to Scandinavia, because Ae is pronounced as A over there) and then ultimately Arenpreis.
The change from -preis to -pries is also relatively easy to explain: the sound of the German -ei- becomes more like "ee" in other Germanic languages, which is written as -i- in Swedish and as -ie- in Limburgish. For example: the German word Preis meaning "prize" is pris in Swedish and pries in Limburgish. Because of this, it made people more likely to spell the second part of the surname as -pries instead of as -preis.
All in all, it looks like the surname might originally have been a German surname that was adapted to one of the Scandinavian languages once your ancestor or one of his own ancestors moved to there. This might partly explain why the surname is so rare and hard to find: it's a unique adaptation of a surname that was already seemingly rare in Germany.
"It is no measure of health to be well-adjusted to a profoundly sick society." ~ Jiddu Krishnamurti (1895-1986)