An interesting research article on this topic is:
Greenberg, Jason H., "From Rochel to Rose and Mendel to Max: First Name Americanization Patterns Among Twentieth-Century Jewish Immigrants to the United States" (2017). CUNY Academic Works.
Comparing names on petitions to the names on passenger lists, the author found the overwhelmingly largest number of females named Ida on naturalization petitions were named Chaya or Khaye on the passenger list. However, there were Ida's also named Chaike, Gute, Hinde, Hodes, Khave, Rode, Rokhl, Taube/Toybe, Yokhved and Yudes. So your best bet is probably to find the passenger list of the ship on which she immigrated.
Richard Gross, Guilford, Connecticut