Tim,
There was a tradition of naming children according to the following rules. Both Scotland & Ireland followed the same system:
The 1st son was usually named after the father's father
The 2nd son was usually named after the mother's father
The 3rd son was usually named after the father
The 4th son was usually named after the father's eldest brother
The 5th son was usually named after the mother's eldest brother
The 1st daughter was usually named after the mother's mother
The 2nd daughter was usually named after the father's mother
The 3rd daughter was usually named after the mother
The 4th daughter was usually named after the mother's eldest sister
The 5th daughter was usually named after the father's eldest sister
The problem with relying too heavily on naming patterns is that, apart from the fact that not all families followed it, there are lots of things that can upset it. Common ones that I have encountered are:
when the father and either of the grandparents have the same name. So you obviously can’t really have two or even three sons all with the same name, so a different name is needed though some families did actually use the same name again on the birth certificate, but differentiated with another more informal name for day to day use. So if you had two Georges (as per birth certs) you just called one of them something else eg Simon;
if a child dies young, then it was common practice to re-use the name for the next birth. And if you don’t know about the death, your analysis of who they were named after could be completely wrong;
Or if a close relative had just died and their name was used out of respect.
I have also noticed that many families liked to name the odd child after the local Minister/Priest, schoolteacher or a relative with no children of their own.
Sometimes people were christened with one name, but went by a different one (and would appear in censuses and other records by that alternative name), eg Henry & Harry or Ann & Nancy.
So for all these reasons, whilst tradition says, for example, that the 4th son was named after the father's eldest brother, it isn’t always so.
Ireland introduced a National School system in the early 1830s but there were other schools before that on an ad hoc basis. Some paid for by the church, some by philanthropic bodies. Many parents couldn’t see the point of sending their children to school and often kept them at home to help around the farm. They considered that more beneficial. So their education was often interrupted during the harvest season or for other urgent work. Most children left school by 10 or 11, with an incomplete education but with some reading and writing skills. The Ordnance Survey memoirs for Donaghmore were compiled in the 1830s and will list the various schools in the parish at that time (and who was funding them). The memoirs may also mention when each school was established.
http://www.nifhs.org/resources/journals/ordnance-survey-memoirs/