Ciebelman or Sibbleman family name is stumping me

My great grandfather, Lazarus Ciebelman, left the Ukraine around 1888 and moved to England, to London's East Side, where many Jewish refugees from European/Russian pogroms were settling then. The only problem is I am stuck trying to find anyone before him because the name Ciebelman/Ciebleman, which I also found as Sibbleman in one of his ship crossings to N. America, is unique, period. No one outside of our family has this name and the name does not exist before Lazarus, making be suspect it is not the original last name. My grandfather Jacob Ciebleman used to tell me with great glee that he came from "Bearshit" or "Chichinik". They sounded funny to me as a kid, but I have assumed that these correspond to real places near where he grew up in (then) Russia, (now) Ukraine: Bershad and Chichilnik, both I believe in the Vinnitsa region of Ukraine.

I would love to have some suggestions as to what to try next.

Dianne