I have researced HEENEY history for about 13 years.My HEENEY family started with John HEENEY who migrated from Navan, Co. Meath.He landed in New Orleans and moved to Louisville KY in 1848.He married Ann KEAGAN in Louisville and in 1859, they and two of their children moved to St. Louis Missouri.They had a son James who is my great-grandfather.
From having done this research, I have found that most HEENEY families came from areas around Dublin in Co. Meath. Some towns in Co. Meath Balbriggin, Navan, Oldcastle and Drogheda with the greatest number being in Drogheda.
If you are researching this name, do searches on Irish cemetery transcriptions and you might have some luck.I would concentrate on County Meath and look for LDS sources on Meath.
I have found that the best way to find a location in Ireland is through an obituary, a church record or a cemetery stone.
Transcribed American Irish cemetery stones are a really good way to locate the county in Ireland- especially if they were buried in New York.