Hi,
I think I've found your Hegedus ancestors in Hungary. I checked familysearch.org and found a couple called Áron Horvát(h) and Eszter Hegedüs in the Reformed Church database. They lived in a village called Gegény in Szabolcs county (Eastern Hungary) and had the following children: Teréz (Theresa) *1877, Erzsébet (Elisabeth) *1879, Veronika *1880, Albert *1881, György (George) *1885, Antal *1886, Juliánna *1887, and Erzsébet *1889. The bad news is that I can't find Barbara (in Hungarian: Borbála). Do you know when she was born?
If she was born after 1889 it could mean that the family moved away from Gegény - perhaps to Budapest. Unfortunately it's very difficult to trace families which moved to the capital city.
There are 21 persons called Borbala Horvath in the Ellis Island database, many of them from Szabolcs county, but only one who arrived in the US in 1913 - she was already 34, so she can't be your ancestor.
You can browse civil registration books of Gegény online from 1895 to 1909 here:
https://familysearch.org/search/image/index#uri=https%3A%2F%...Gyula Hegedüs
Budapest