As thoraway pointed out, Øie/Øye is not actually part of the name, it is more like an address. Every time they move to a different farm, this changes accordingly.
The church confirmation record helps show this:
Confirmed 5 Oct 1879
Anna Bertine Pedersdatter
Born 11 Aug 1865 at Øie
Baptized 3 Sep 1865
Presently living at Haugeland
Father: Peder Hanssen
Mother: Anne Kristoffersdatter
Kildeinformasjon: Vest-Agder fylke, Feda i Kvinesdal, Ministerialbok nr. A 3 (1877-1882), Konfirmerte 1879, side 41.
Permanent sidelenke: http://www.arkivverket.no/URN:kb_read?idx_kildeid=6210&i...
As you can see, somewhere between 1865 and 1879 the family moved from the farm "Øie (Øye øvre)" to "Haugeland".
Øye is more of a rural community than it is a farm. It has two main parcels, Øye øvre and Øye yttre. In the 1865 census there are 278 people living in dozens of families on Øye.