Thank you for your replies. I will check into the Hart entry in Griffiths. I did modify my post to correct the birth date and to add some names.
Our biggest problem in this research has been determining the town. Some family history says Meath, Enfield (Meath), or Dublin - even "19 miles NW of Dublin". There is one reference to Cork from someone who signed a death certificate.
Enfield is the latest and most specific info we have from a descendent of Michael Jones.
We know Thomas Francis Jones arrived in New York in 1860 at age 18. He brought his mother, Mary Fagan Jones Hart, sister Mary Jones, and step-father Owen Hart (wtih sons Patrick, James, Bryan, and John) to the US in 1865. All sailed from Liverpool.
I guess I'm hoping by linking some combination of Jones, Fagan, and Hart in the same locale, it will narrow down the town and ultimately lead to the parish. I know this is a long shot. It is quite different than the research I have done here in the US.
Thanks again for your help!