The tithe applotment records list Andrew Acheson farming in Ballyligpatrick in 1825. That’s on the modern Buckna Rd 2 or 3 miles east of Broughshane. There’s a Presbyterian church and graveyard at Buckna. Baptisms and marriages start in 1836 but there’s an index to baptisms from 1828. Presumably the early records were lost. (Copies held in PRONI). The church was established in 1756. Presbyterians in Ireland generally don’t keep burial records.
PRONI also has a record of a will being probated for Andrew Aitcheson of Ballyligpatrick in 1839, In the Pre 1858 Wills and Admons (Connor Diocesan Wills). The will itself was destroyed in the 1922 fire in Dublin. All that remains now is an index book with his name and the year.
There are no Achesons in Ballyligpatrick in Griffiths Valuation c 1860 so they had either died out or moved away by then. Population in 1901 was 209. Practically every person there was Presbyterian. (So descended from Scots settlers who mostly arrived in the 1600s).
The Acheson farm was probably rented from Lord O’Neill of Shane’s Castle near Randalstown. His estate has some rental books which I have used previously. They are now in PRONI. Dating to 1828 T1024/1 & 1832 T1024/2 they should tell you the size of the farm, the terms of the lease and may include names of earlier family members. A lot of the farms on that estate had 3 lives leases granted in the 1780s and 1790s, and some of those leases gave names and ages for the 3 lives. The records are not on-line. They are in a couple of books. Personal visit required to view them.