I started working on this a few weeks ago but then got caught up in a whirlwind on a branch of my mom's family so I set it aside. I'm not 100% convinced yet but there are a lot of coincidences.
First, what I do know:
After making contact with the family of Joseph Hochfeld (son of Benjamin and Elka...Elka is the sister of my gg-grandfather) I was working on the Hochfeld name and came across something interesting on Elka Hochfeld's (nee Belinki) 1946 death certificate. Benjamin reported her as being born in Poland. Considering the entire family listed Zwenigorodka, Russia (Ukraine) as their last residence when coming to the US and Benjamin and Elka are buried in a Zwenigorodker Society section, this reference to Poland stands out. Elka's parents are listed as Gershon Belinki and Chaia and based on her age on passengers lists and census records her date of birth was circa 1865. My g-grandmother Lea Belinki and her brother Leib Belinki listed Lysyanka, Ukraine as their last residence and place of birth on their ship manifests. By 1913 when their brother Shlomo Belinki came to the US, the remaining family was living in Talne, Ukraine. My Ukraine researcher couldn't find any history of the Belinki name in Lysyianka or Zwenigorodka.
Second, my theory:
I went to the JRI-Poland website and searched for "sounds like" Belinki surname. There were very few hits on names that closely resembled Belinki so I tried searching combinations of given names. I tried a "sounds like" given name Elka + "sounds like" given name Gershon. In the town of Suwalki, I found records for Berlinski/Berlinska including Elka born 1865 to parents Abram Hirsz and Chaja Fejga. Because of the "R", Berlinski is a different soundex than Belinki. Gershon and Hirsz are not exactly the same but since people with 2 given names tended to use them interchangeably it is a possible match. Here are the Suwalki records I found:
>Abram Hirsz b. 1832 (son of Lejzor and Chaja (daughter of Ablowna)). In 1850, he married Chaja Fejga Filipowska b. 1833 (daughter of Eliasz/Elka and Chana/Chaja).
>>Sora Estera b 1853, daughter of Abram Hirsz and Chaja. She married Abram Turtulski in 1878.
>>Lejzor b. 1856, son of Abram Hirsz and Chaja.
>>Idzko b. 1859, son of Abram Hirsz and Chaja.
>>Brajna b. 1865 (twin), daughter of Abram Hirsz and Chaja. She married Litman Gersztinger in 1881.
>>Elka b. 1865 (twin), daughter of Abram Hirsz and Chaja...future wife of Benjamin Hochfeld??
I don't see any more birth records after Elka. My gg-grandfather Abram Belinki may have been younger than Elka. Her oldest child was born in 1885 and I think my g-grandmother Lea, who I believe is Abram's oldest, was born c. 1889. Either there's a gap in the records (doesn't look like it though), the birth wasn't registered in Suwalki for some reason, or if this is my family, maybe they were already moving to the towns in Ukraine that I've been seeing and Abram was born there and not registered (or records didn't survive or just aren't indexed). A man named Abram Hirsz would not name a son Abram unless the son was being named for someone else or the father could have died while mom was pregnant so this may be one point against my theory.
Two Turtulski children (Rochel and Elli), likely children of Sora Estera based on their ages, arrived in the US in 1904 to see uncle L. Berlinski in Washington DC, likely Lejzor. By 1940, Elli Turtulski applied for citizenship in Miami, FL. under that name with an "also known as" name Aleck Stein.
When I go to NY for Passover next year I want to photograph the Hochfeld headstones at Mount Carmel Cemetery. It would be amazing if Elka's father is listed as Abram Hirsz. A birth record for my gg-grandfather Abram would have really made my day. I'm familiar with the Suwalki gubernia records through research on my mom's side of the family tree. Poland is always easier to research than Ukraine so that would be nice.
The names Belinki and Bilinki do not appear in A Dictionary of Jewish Surnames from the Kingdom of Poland. The name Berlinski does appear here. The names Belinkij and Belinskij (the family used Belinsky in the US) are listed in A Dictionary of Jewish Surnames from the Russian Empire. The name Berlinskij is also listed here. The names have different meanings...Berlinski means from the village of Berlintsy (Mogilev-Podinskiyy, Ukraine). Belinki means white or blonde.
Update: I discovered that Brajna and Elka are not twins, their births were just registered at the same time. Elka was born in 1864 and Brajna is a little older. Still a possibility but no connection proven or disproven yet.