Rob,
Re-using a child’s name when the first one died was very common. Families liked to keep names alive (literally) and so what you noticed happened a lot. It doesn’t mean it was specially important and was the father’s father (though it might have been). It was simply a very common thing to do.
Orr was a common surname as you will know. If your family were farmers then the head of household would normally be in the tithe applotment records which were compiled around 1828/1834 for Armagh. So this gives you a list of parishes and townlands where Orrs lived at the time you are interested in.
Your best bet of finding them is from church baptism records. However, as you probably know by now, not all churches have records for the years you need. (40% of all the early Church of Ireland records in the country were destroyed in the 1922 fire. There’s no easy way round that). It would also help to know precisely what denomination they were. Protestant includes Church of Ireland, Presbyterian, Baptist, Methodist and some others. Each kept separate records. Most are not on-line, though the most complete set is in PRONI. Going through all the possible records for Armagh would be quite a daunting task.
Looking at the 1901 census of Armagh, there were about 150 Orrs. They were fairly evenly split between Presbyterian, Church of Ireland & Methodist. (Methodism was slow to become established as a separate denomination in Ireland and it was about 1820 before they started baptising and the 1830s before they began their own marriages. Most continued to use the Church of Ireland until much later. So if searching church records I would focus on Church of Ireland & Presbyterian.
Few churches have records for 1770. If your William was born then, there’s not a great likelihood you’ll find him. There’s a general lack of records in Ireland for the 1700s and early 1800s and so not many other avenues to suggest.
Orr, Anne-- Townland : Drumnasoo Year : 1828-Loughgall-Armagh
Orr, Archibald-- Townland : Drumnasoo Year : 1828-Loughgall-Armagh
Orr, Catherine-- Townland: Ballynick Year: 1832-Grange-Armagh
Orr, David-- Townland : Marleycoo More Year: 1834 -Mullaghbrack-Armagh
Orr, David-- Townland: Marleycoo More Year: 1834-Kilclooney-Armagh
Orr, Hugh-- Townland: Ballyloughan Year: 1833-Kilmore-Armagh
Orr, Jacob-- Townland : Ballymagerny Year : 1828-Loughgall-Armagh
Orr, James-- Townland: Annarea Year: 1833-Kilmore-Armagh
Orr, James-- Townland: Ballynick Year: 1832-Grange-Armagh
Orr, James-- Townland: Greenan Year: 1832-Grange-Armagh
Orr, Jas. T: Derrycreevey Knox Y: 1833--Clonfeacle-Armagh
Orr, Jas. T: Stangmore (Knox) Y: 1833--Clonfeacle-Armagh
Orr, James-- Townland: Greenan Year: 1832-Grange-Armagh
Orr, James-Townland: Belteagh Year: 1827-Tynan-Armagh
Orr, Jas. T: Derrycreevey Knox Y: 1833--Clonfeacle-Armagh
Orr, Jas. T: Stangmore (Knox) Y: 1833--Clonfeacle-Armagh
Orr, John-- Townland : Ballymagerny Year : 1828-Loughgall-Armagh
Orr, John-- Townland : Derrycrew Year : 1828-Loughgall-Armagh
Orr, John-- Townland : Teemore Year: 1834 -Mullaghbrack-Armagh
Orr, John-- Townland: Ballynick Year: 1832-Grange-Armagh
Orr, John-- Townland: Teemore Year: 1834-Kilclooney-Armagh
Orr, John-County: Down Townland: Ballyinagin Year: 1834-Magheralin-Armagh
Orr, John-Townland: Lurgan Town Year: 1833-Shankill-Armagh
Orr, John-Townland: Tegnavin Year: 1833-Shankill-Armagh
Orr, John-Townland: Tollygalla Year: 1833-Shankill-Armagh
Orr, Joseph-- Townland: Ballyloughan Year: 1833-Kilmore-Armagh
Orr, Joseph-Townland: Cranagill Year: 1834-Tartaraghan-Armagh
Orr, Mr. - Townland: Mecantrim Year: 1834--Kilclooney-Armagh
Orr, Mrs. - Townland: Ballyloughan Year: 1833--Kilmore-Armagh
Orr, Rob. T: Coolhill Y: 1833--Clonfeacle-Armagh
Orr, Robert T: Lisnacroy Y: 1833--Clonfeacle-Armagh
Orr, Robert-- Townland : Ardress East Year : 1828-Loughgall-Armagh
Orr, Robert-- Townland : Ballymagerny Year : 1828-Loughgall-Armagh
Orr, Robert-- Townland : Coragh Year : 1828-Loughgall-Armagh
Orr, Robert-- Townland : Teemore Year: 1834 -Mullaghbrack-Armagh
Orr, Robert-- Townland: Ballyloughan Year: 1833-Kilmore-Armagh
Orr, Robert-- Townland: Teemore Year: 1834-Kilclooney-Armagh
Orr, Robert, Jr.-- Townland : Drumnasoo Year : 1828-Loughgall-Armagh
Orr, Robert, Sr.-- Townland : Drumnasoo Year : 1828-Loughgall-Armagh
Orr, Robt. T: Annagh Y: 1833--Clonfeacle-Armagh
Orr, Robt. T: Roan Y: 1833--Clonfeacle-Armagh
Orr, Robt., Sr. T: Kilnagrew Y: 1833--Clonfeacle-Armagh
Orr, Simonn-- Townland : Markethill Year: 1834 -Mullaghbrack-Armagh
Orr, Simonn-- Townland: Markethill Year: 1834-Kilclooney-Armagh
Orr, Thomas-- Townland : Demesne of Hamilton Bawn Year: 1834 -Mullaghbrack-Armagh
Orr, Thomas-- Townland: Demesne of Hamilton Bawn Year: 1834-Kilclooney-Armagh
Orr, William-- Townland : Anaghmore Year : 1828-Loughgall-Armagh
Orr, William-- Townland : Ballymagerny Year : 1828-Loughgall-Armagh
Orr, William-- Townland : Coragh Year : 1828-Loughgall-Armagh
Orr, William-- Townland : Derrycuhan Year: 1834 -Mullaghbrack-Armagh
Orr, William-- Townland: Ballynick Year: 1832-Grange-Armagh
Orr, William-- Townland: Derrycuhan Year: 1834-Kilclooney-Armagh
Orr, Willm. T: Derrygoonan Y: 1833--Clonfeacle-Armagh