You could search for William Gibson in the tithes applotment records for co Down , which are incomplete and worse is many were conducted in 1834-5 and contain several William Gibson
http://www.irishgenealogyhub.com/county-down-genealogy.php#.Vg0WxUvU4YUCounty Down Tithe Applotment Records 1823 to 1838 (by Civil Parish)
There are 69 civil parishes to search:-
some examples
Comber The date collected is 1835 so too late
Donaghadee The date collected is 1834 do look up
Inch. do look up
Gibson, Aaron Townland: Ballygawley Year: 1825
Gibson, Aaron Townland: Ballygawley Year: 1826
Gibson, John Townland: Ballygawley Year: 1825
Gibson, John Townland: Ballygawley Year: 1826
Gibson, Saml. Townland: Ballygawley Year: 1825
Gibson, Samuel Townland: Ballygawley Year: 1826
Gibson, William Townland: Ballygawley Year: 1825
Gibson, William Townland: Ballygawley Year: 1826
Kilmegan
Gibson, David Townland: Castlewellan Year: 1829
etc etc but a systematic search of all is an important negative or positive search of one of the few irish records covering the date before emigration
other sources include
http://www.bagenalscastle.com/ancestry/index.htmNewry Town Agricultural Census 1803
Firstname Surname Location
William Gibson Drumadonald
http://www.proni.gov.uk in particular
http://apps.proni.gov.uk/freeholdersThere are over 80 entries for William Gibson, the dates on many are in the right period
Gibson William Ballygally D/654/A3/1A More Details .
Gibson William Ballymoneyally D/654/A3/1A More Details .
Gibson William Bangor D/654/A3/1A More Details .
Gibson William Drumalig D/654/A3/1A More Details .
Gibson William Ballydugan D/654/A3/1A More Details .
Gibson William Ballyworfy D/654/A3/1A More Details .
Gibson William Ballyworfy D/654/A3/1A More Details .
Gibson William Toghblane D/654/A3/1A More Details .
Gibson William Ravarna D/654/A3/1A More Details .
Gibson William Corcreeny D/654/A3/1A More Details 1788
Good luck onwards Ballyaltikilligan G