I'm confused why you are posting on the Romania Board if your family is from Budapest (Hungary)?
Veroninka, Étel, and András are on the Ellis Island database (August, 1902), but their last residence does not show as Budapest. On the manifest, it looks like "Lieszek". If so, this town is currently in Slovakia, though formerly in Árva count of the old Hungary.
I don't see microfilmed church records of Lieszek itself, though there are some for the nearby village of Csimhova.
The family is going to János (John) Tocsik in Wilmington, Delaware.
I would suggest posting on the Slovakia or Hungary boards.