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This is a really interesting name for me, mainly because I suspect that there are several languages at work. There's a lot of guesswork involved, so take nothing for granted.
I'm pretty sure, that this is the same as the Slavic name Berestiansky, which is either Ukrainian or Bielorussian, or both. This surname derives from a place-name who's form I can't reconstruct, but which certainly incorporates the Slavic word for a birch tree.
The -er ending shows that this form developed in a German- or Yiddish-speaking community. My guess is Yiddish.
Then there's the letter j, not to be expected for a y sound in a Yiddish name. It's possible that this spelling appears under the influence of the Polish language. The whole surname seems to reflect your ancestor's "lot of travelling." But remember, most of this is speculation.