Although it sounds an English Name, Cunningham is actually Scots. Google says "
This distinguished surname recorded in the spellings of Cunningham, and the rarer Irish Conningham and Conaghan, is of early medieval Scottish origin. It is a locational regional name from Cunninghame, one of the territorial divisions of the county of Ayrshire.
www.surnamedb.com/Surname/Cunningham "
It is however not Gaelic in origin, Ayr is in southern Scotland and therefore in the area of Anglian settlement in Early Mediaeval times. However a lot of Cunninghams do live in the former Gael settled areas, as well as in the Northern Isles (Phil Cunningham the fiddler is from the Shetlands.)
There's no short jumps for you I'm afraid; you'll just have to do a careful search back to find your family start-ups!
Good Luck,
Chris B.