My great-great grandmother Ida Zacharowicz was born in Russia and emigrated to the U.S. in 1890. Ida was NOT too anglicized to be given to a Jewish girl! In fact Ida was quite popular for Ashkenazi girls at the time. Below are a few of the famous Jewish Idas from the late 19th early 20th century, so it stands that there were even more non-famous ones!
Ida Cohen Rosenthal, co-founder of Maidenform, b. 1886 in Rakov, Russia
Ida Kaminska, actress known as the "Mother of the Jewish stage", b. 1899 Ukraine
Ida Maze, the Yiddish writer, b. 1893 in Ugli, Belarus
Ida Demel, writer and patroness of the arts, b. 1870 in Germany
Ida Rubenstein , ballet dancer, b. 1883 in Ukraine
Ida Haendel, pianist, b. 1928, Poland
Ida Ginsburg ,founder of the Detroit chapter of National Council of Jewish Women, b. 1865, USA
As for the corresponding Hebrew name, I would guess Yehudit, or Adah.
BLUM, ZACHAROWICZ, SACHS/SACKS, SPEICHLER, ROTHBERG, KIER/KIERSCH