Stranocum (town and Townland) is 1.7 miles southeast of the Dervock Presbyterian Church as the crow flies and Ballyrobin Townland is adjacent to the west of Stranocum Townland. I wonder if the subject Archibald's father could be this individual (who is perhaps the Archibald Gardner noted in the 1803 Agricultural Census for Stranocum Townland or perhaps the person in the census is the grandfather):
Archy (Archibald) Gardner/Gardiner, shoemaker, Cuppidale, Stranocum, Ballymoney District, County Antrim, Ireland, daughter Margaret Gardiner, spinster, maid, witness Ella Lilley, m. 04 JUN 1863, Robert Campbell, bachelor, laborer, father William Campbell, shoemaker, Cuppidale, witness Archibald Lilley, married by John Beatty, Roseyards Presbyterian Church, Parish of Ballymoney, Ballymoney District, County Antrim, Ireland (
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1863/09643/5497128.pdf)
If so, possible relations (uncle or great uncle?) are:
Alexander Gardiner, one spinning wheel, Ballymoney Parish, Flax Grower's List North Antrim 1796 (
http://www.billmacafee.com/1796flaxgrowerslist/1796flaxgrowersantrim.pdf).
Alexander Grardner, Ballyrobbin (Ballyrobin) Townland, 1803 (
http://www.billmacafee.com/1803agriculturalcensus/1803censusantrim2.pdf).
Alex-r Gardner, Ballyrobbin (Ballyrobin) Townland, 1825, adult “occupying” agricultural (not urban) land at this time (Ireland Title Applotment Books 1825, Ballyrobbin Townland,
https://www.nidirect.gov.uk/services/search-pronis-ecatalogue, search the eCatalogue, FIN/5, Browse, FIN/5/A, Search, or navigate to FIN/5/A/40 for Ballymoney Parish, could not get .pdf files to open with Adobe but would open in Chrome;
http://www.irishgenealogyhub.com/antrim/tithe-applotments/ballymoney-parish.php, transcribed index).
