Is this the town?
It seems this is a hamlet in the region of Minsk. Currently about 138 residents.
I am also searching a very tiny "town" that today is in Poland (Daliowa / Daljowa.) I have found it impossible to find any records for my tiny town or the people that lived there.
The only thing I can suggest is to try searching at The JewishGen Unified Databases for each variation of the town name. Byenitsa, Bienica, and Benitsa
Do the same at databases such as www.familysearch.org and Ancestry.com
You can put the town name in the birth field and leave everything else blank. That's something that works well with small towns.
RATOWSKY / CHAIMSON (Ariogala, Lith.)
LEFFENFELD / FINK / KALTER (Daliowa & Jasliska, Pol.)
BOJDA / BLEIWEISS (Tarnow & Tarnobrzeg, Pol.)
WALTZMAN / WALZMAN (Ustrzyki Dolne, Pol.)
LEVY (Tyrawa Woloska, Pol.)
SOLON / SOLAN / SOKOLSKY (Grodek, Bialystok, Pol.)
BOBISH / BLUMENKRANZ / APPEL / WEINER / ROSENBERG (Vysoko-Litovsk, Brest, Biala Podlaska)