I think the townland where the family lived was probably also known as Carreyreagh, as well as Canneyreagh and was/is 261 acres of agricultural land. It is a mile or so west of Donaghadee village and is in the parish of Donaghadee.
In Griffiths Valuation (1863) there were 18 households there. (No Nevin or McMinn). It is listed as Carreyreagh in Griffiths. The modern Canneyreagh Rd runs through it today.
http://www.askaboutireland.ie/griffith-valuation/index.xml
Donaghadee Church of Ireland records start in 1771. For Presbyterians, there’s a choice of about 10 churches in the area. (Which you attended was a matter of personal preference and not always the closest). Donaghadee 1st Presbyterian’s records start in 1793. There’s a copy of all the surviving church records for that area in PRONI (the public record office) in Belfast. The records themselves are not on-line and a personal visit is required to view them there.
The tithe applotment records for 1834 list a B. C Nevin Esq. in Cannyreagh. He also appears in Ballymacruise townland and so presumably also had land there.
http://www.irishgenealogyhub.com/down/tithe-applotment-books/donaghadee-parish.php#.Vqm_UVJ2ug0
Esquire means he was of some standing in the community and probably of independent means. There were some McMinns nearby in Herdstown & Hogstown, again Esq. So that might be Jemima’s family.
B. C Nevin is probably Benjamin Chapman Nevin. PRONI have a mortgage in his name, on property in Ballymacruise, which indicates he was living in Paris (France) in October 1832. D366/492.
See also D664/DA/691, which is described as “George Dunbar's copy of a Landed Estates Court rental and particulars of sale of lands in Ballymacruise and Carrowreagh, Co. Down” Benjamin Nevis (spelling) was listed as the owner. The involvement of the Landed Estates Court usually indicates the landlord was in some financial difficulty and couldn’t settle his debts.
See also T2265 which was: “Encumbered estate court rental, with maps and particulars, relating to the sale of Benjamin Nevin's estate at Ballymacruise and Carrowreagh, Co. Down”. Dated 1864.
Source: http://www.proni.gov.uk/index/search_the_archives/ecatalogue.htm
These records are not on-line and, if you can’t go in person, you’ll need a researcher to get them for you. (Or pay PRONI a hefty fee to copy them).
I also noted this probate abstract on the PRONI website:
Nevin, Benjamin of 4 Bensham Avenue Gateshead 8 county Durham died 27 July 1965 Probate Newcastle-upon-Tyne 12 November to Andrew Galbraith solicitor and George Magnay chartered accountant. Effects in Northern Ireland £158 8s. Re-sealed Belfast 23 December.
Benjamin Nevin is quite an unusual name and although this relates to someone who died in 1965, I’d wonder if there’s a family connection. The full file should be held in PRONI. (Re-sealed means he had property in more than one jurisdiction presumably England & Northern Ireland).