Virginia,
Researching in Ireland in the 1700s is very hard going due to the general lack of records. You need to rely heavily on church baptism & marriage records but not all churches have records for the 1700s. You also need to know the family's precise denomination. That said, for emigrants in the mid 1700s, the vast majority were Presbyterian or Church of Ireland (ie Anglican).
Looking at Griffiths VAluation for Co. Armagh in 1864, there were just 3 Hoy households listed. In the parishes of Seagoe and Drumcree.
In the 1901 cenus there were 64 Hoys in the county, a mix of Church of Ireland, Presbyterian and RC.
http://www.census.nationalarchives.ie/search/
You could get a researcher to search Church of Ireland and Presbyterian records for Seagoe & Drumcree. Those are held in PRONI in Belfast. Otherwise DNA testing might be another way of possibly finding a match with someone who knows more about the family origins.
In my experience, given the name, location and likely denomination, the family is probably descended from folk who settled in Armagh from Scotland or England in the 1600s.