After Elizabeth dies, there is a niece with the family on the 1920 census, Anna Goeb. I thought this might be an avenue of exploration but... I'm not getting anywhere:
1920 census - "United States Census, 1920," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:MDT3-4X2 : accessed 7 August 2018), Elizabeth Amon in household of Gustav Amon, Perry, Franklin, Ohio, United States; citing ED 294, sheet 14A, line 39, family 176, NARA microfilm publication T625 (Washington D.C.: National Archives and Records Administration, 1992), roll 1380; FHL microfilm 1,821,380.
1910 census, her parents are listed as being born in Germany, but the Amons were also listed as being born in Germany at this time too (the Austian Hungarian Empire was very large and complicated and was only dissolved in 1918, which is why the 1920 census typically shows true countries of birth, instead of Austria, Hungary or Germany).
"United States Census, 1920," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:MDT3-4X2 : accessed 7 August 2018), Elizabeth Amon in household of Gustav Amon, Perry, Franklin, Ohio, United States; citing ED 294, sheet 14A, line 39, family 176, NARA microfilm publication T625 (Washington D.C.: National Archives and Records Administration, 1992), roll 1380; FHL microfilm 1,821,380.
Anna's possible birth reference, although how helpful it is at this point is yet to be known. If she is a niece of Gustav Amon but had a different name to him I had hoped that her mother was his sister, but the name is different again!
"Ohio, County Births, 1841-2003," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:VRM2-9BW : 13 March 2018), Anna Elizabeth Gabe, 20 Aug 1902; citing Birth, Newark, Licking, Ohio, United States, county courthouses, Ohio; FHL microfilm 384,321.