From Ruth Marcus, November 18, 2002:
My g-g-g-grandfather was Rabbi Yosef Eliashberg, son of R. Yehuda-Leib. R. Yosef was born around 1799 in Rumsiskes (Rumshishok), Kovno Gubernia, and was also known as Yosef Rumsisker. He worked as a Moreh Tsedek (teacher of Torah) to youngsters in Rumsiskes. When R. Yosef became old, he left Lithuania and came to Eretz Israel. He died in 1870 and was buried at the cemetery on Mount of Olives in Jerusalem.
Remark: It is possible that R. Yosef Eliashberg was related to the Eliasberg rabbis of Ivenets, Minsk Gubernia. However, at this point we cannot prove nor disprove the existence of such a relationship.
R. Yosef and his wife Yocheved had a son named Mordechai (they had more children, but their names are unknown). Mordechai was born in 1816 in Cekiskes (Chikishok), Kovno Gubernia. In 1826 R. Yosef and his son Mordechai went to Slobodka (near Kovno) and stayed at the house of Markil Kadishon. In 1830 Mordechai married Ester, a daughter of Markil Kadishon. From 1866 until 1899, R. Mordechai Eliashberg served as a chief Rabbi in Bauska, Courland Gubernia (nowadays in Latvia). He was among the pioneering messengers of the Orthodox Zionist movement that was established formally about 12 years after his death. R. Mordechai was in a close connection with Rabbi Abraham Itzchak HaCohen Kook who served as a rabbi in Bauska from 1895 to 1904, before his arrival to Eretz Israel. R. Mordechai died on November 29 of 1889 in Bauska, aged 73, buried at the Jewish cemetery in Bauska. There are streets in Tel Aviv and in Jerusalem named after Rabbi Mordechai Eliashberg.
R. Yosef and R. Mordechai Eliashberg have mentions in the book Yahadut Lita, Vol. 3 (Hebrew) as well as in other sources. In addition, R. Mordechai has an item in The Hebrew Encyclopedia, Vol. 3, page 534.