Alexander Hamilton, born 1705 in Down, Down Ireland

There are very few records in Ireland for the 1700s. And as you say, it’s a very common name. Looking at the 1901 census there were 119 people named Alexander Hamilton in Ireland at that time. You would need to have your Alexander’s parents names to have any certainty of identifying the right family, but the main obstacle is that very few churches in Ireland have any records for a birth in the early 1700s. There may not been any documentary record of this family. DNA testing might help you locate someone alive today who is from the family but otherwise I doubt there is much prospect.

The name suggests a Plantation family. The family probably had only been living in Ireland for a short period (after coming from Scotland) and so that won’t help locate them either.

He is as far as I can go back in this line at this time.  He died in Armagh. His children were born in Dublin as was his wife. Eventually, one of the sons came to America, married another Irish girl and they all go on from there. Thank you for your help.