Sonja,
School attendance records are not on-line. PRONI hold most of the surviving records but they are in paper format and you have to look them up there in person. The following attendance records exist for schools in the parish of Coleraine:
Bohill National/PE School Antrim SCH/816
Coleraine Academical Institution Londonderry SCH/612
St Malachy's National/PE/PS Londonderry SCH/1194
St John's National/PE/PS Londonderry SCH/1352
Coleraine Model/PE/PS Londonderry SCH/422
The Honourable Irish Society's Coleraine Free Londonderry SCH/529
Blagh Public Elementary School Londonderry SCH/820
For Killowen:
Waterside Girls's National/PE/PS Londonderry SCH/1303
Waterside Boys's National/PE/PS Londonderry SCH/1364
I don’t know what years the records cover. You would need to look each set up to find out.
I think you can reassure your husband that his belief that the family are of Scottish origins is 99% likely to be correct. Most of the entire population of Coleraine and the surrounding area at that time were of Scottish origin, so it would quite surprising if they didn’t come from Scotland. Possibly worth explaining that at the nearest point it’s only about 10 miles across to Scotland. (The modern ferry from Ballycastle to Campbeltown takes 40 minutes). It wasn’t the biggest journey they had to make. (See attached photo of the Scottish coast taken from the Giant's Cuseway area).
However most arrived in the 1600s. By the 1700s they were starting to leave again. Gabriel will have been born in the general Coleraine area I would expect. However you will struggle to find any proof of that as the records you need simply don't exist. The connection back to Scotland will be 100 – 150 years before his birth.
Speaking of the district around Killaig and Ballycahan (near Coleraine) in the 1820s, Dr Bruce the local Minister said: “Over a space of 15 to 20 miles from east to west and about the same from north to south, Scottish surnames, a broad Scottish dialect, and an almost universally diffused Presbyterianism indicated the title of the people to call themselves “Scotch.” Episcopalian protestants ie Church of Ireland were few and a Roman Catholic was almost as rare as a black swan.”
P13 of “A Kennedy Chronicle - Alexander Kennedy of Ballycahan”, by Hugh Alexander Hezlettt.
Speaking of his youth in Aghadowey in the 1830s, page 25 of Autobiography of Thomas Witherow 1824 – 1890, pub 1990 by the Ballynacree Historical Society:
“Aghadowey had originally been settled by a Scotch migration and I found that my new neighbours spoke as pure Scotch as a man might hear in any part of Ayrshire.”
Even today the accent in parts of Ulster is essentially Scottish. Quite different from accents you may hear elsewhere in Ireland. For a general description of the background that brought 200,000 Scots to live in that part of Ireland in the 1600s, you might want to read “Eagles Wings - the Journey of the Ulster-Scots & Scots-Irish” by Dr David Hume.
For a general description of life in Coleraine in the 1700s, you might want to read “Coleraine in Georgian Times” (1977) by Rev Mullin. (There’s a copy in Coleraine Library, local studies section). You should try and get to Coleraine Library to see what they have in that local studies section. You could e-mail the library (when it re-opens) and ask the Local Studies staff for their help. They may hold copies of family histories etc and other local reference books that might mention Gabriel.