My husband's mother was a Tyne, from the Tyne family of Tipperary, specifically Gortnahoe Parish (aka New Birmingham/Thurles area).
My husband's great-great grandfather is John Tyne (born ca1820 @Foilecamen; died 1891@Oldtown, Co.Tipperary--tombstone @Sacred Heart Cemetery, Gortnahoe). Our John Tyne was married to Ellen Lee (1834-1905) on 20 Nov 1851 at Gortnahoe, according to diocese records of Cashel & Emly.
I am attaching a scan of the original Gortnahoe parish records for 1851. Their record is "20" under November.Here's my transcription of those 2 lines: "20. John Tyne & Ellen Lee of Oldtown were married in presence of John & Judith Lee" followed by scribbled signature of the priest? recorder?
Because of your connection to John Tyne & Ellen Lee, I am quite certain that your great-grandmother's parents are also the parents of Edmond Tyne, who emigrated to New York City in 1880 (oral records from USA Tyne family). That family line produced my husband's mother within 2 generations (by 1920s). This branch has remained in the NYC and New Jersey areas ever since.
I recently discovered that part of the family also emigrated to and settled in Nashville TN.
James Tyne & Catherine Healy (married @Ballingarry, Tipperary 1864) settled in Nashville TN (1870 US census), where James died sometime before 1880. A few records name him as William James Tyne; some James W. Tyne. He seems to stick with James W. in the US. His son Thomas J. Tyne became a wealthy prominent citizen of Nashville during the early 20th century. I don't know about the O'Neill hotel, but James Tyne and his family were settled there and James was involved in constructing the first railroad bridge that crossed the Cumberland River at Nashville. Newspaper records also show that he was involved in real estate transactions. At that time in that part of the US, one could make a lot of money on land speculation and development. Perhaps James's sister Mary also had a part in their going to Nashville to try to make a living as her brother was doing well there.
As you have been able to trace one of the daughters of John & Ellen (Lee) Tyne, congratulations! I would like to know more about the daughter Mary. What I have is nothing more than her first name. John & Ellen (Lee) Tyne had 5 sons and 4 daughters according to my family source. Information on any of the daughters is scant.
I can only give suggestions about emigrating to the US as that has been my focus. There are U.S. records before Ellis Island. In general this site is a good place to look: http://www.archives.gov/research/immigration/passenger-arrival.html#col…. This is also good (part of same NARA site): http://www.archives.gov/research/immigration/#where.
https://aad.archives.gov/aad/series-list.jsp?cat=GP44 is the page on NARA's site where you will find a searchable database specifically for Irish immigrants to NYC during the famine.
It is often helpful to know which port or point of entry was used to get into the US as there were many. Most, however, arrived through the port of New York.
If any of this is useful, happy hunting!