John Jack Black Oneill

The name you have might be explained as the result of common naming customs. Often, in areas where there were a lot of people with the same surname and a limited number of customary given names, people would (and still do) differentiate among them by reference to their father's names or some physical characteristic or other attribute

For example, a relative of mine named John, living in an area where many people had the same surname, was called John Jack Ricky, because his father was also a John (who was called Jack), and his grandfather was named Richard/Ricky.

In other cases, someone might be called Seán óg, or young Seán, if he or someone else nearby had the same name, but the other one was older or bigger.
Another common way of distinguishing people from one another was hair color, so Seán Rua would be red-haired Seán, and Seán Dubh would be dark-haired Seán. The word Black in the name you have may have been Dubh originally, and simply got translated when talking about him after emigration.
So, your ancestor may have been John, son of black-haired Jack.

Hi Both

Thanks so much for your responses, thats so helpful and appreciated. Looks like testing is the way forward.

Thanks again

I forgot to mention a source who might be able to help you in your search.

As you probably know already, the surname O'Neill originated in Ulster, but after the Plantation of Ulster in the 17th century, a number of O'Neill's migrated to other parts of Ireland.  Some went to Mayo, where two of my grandparents grew up, and I have some distant O'Neill relatives there.  There is an O’Neill heritage project headed by Sean O'Neill, who is from the O’Neill branch in Mayo, and whose email address is sean@oneill.ie.  His family comes from near Swineford in East Mayo, but his research covers other parts of Ireland as well, so he might be able to help you locate your ancestors.

In the old days, it was customary to call your children after your father or grandfather. Many of our ancestors inherited a nickname, because too many people had the same name. The facial features, the skin color or the hair color directly influenced the nickname. This is the case for one of my ancestors John Quane Black and Maurice Dineen who inherited this nickname for his black hair and skin. I find it crazy to discover all this, to understand how much genetics influences history, even the names. I read an article recently that explained how our hair gets these properties.