Verily, have you had any chance to look through Irish church records? My wife says that once you get them traced back to Ireland that's all you can use.
Ah, but it's the getting to Ireland that's the trouble. And there are too many fecking Byrnes and O'Byrnes and various alternative spellings in Ireland to be making guesses about Peter Byrne's parents or grandparents. (Like I said, we're not even sure they were Irish.)
Well Byrne is one of the ancient Royal families in Ireland and quintessentially Irish. Further Bridget was such a stereotypically Irish name that servant girls were called Biddies in the old slang. Heaton while not of direct Irish origin was found in many parts of Ireland. The English often tended to anglicize an Irish name to its closest English equivalent. Ill have to face it that your likely part paddy.
If it is any use- there was a Bridget Heaton who arrived into NYC as an infant in 1852 from Liverpool by way of Ireland.
Another Bridget Heaton arrived from Ireland into NYC in 1865 at the age of 15.
Lastly there was another Bridget Heaton who arrived in 1849 likely Ireland via Liverpool born 1835. Let me know if any of these fit a description and I could probably email or pm you the manifest. Hope you dont mind the prying- I pay for a subscription so I like to get some use out of the site every now and then