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It is a Scottish surname but my information is that it derives from an English place name, Conisborough in Yorkshire.It's an old surname in Scotland; first on record, Galfridus (Geoffrey) de Coningesburg in 1164. Cuniburgh is a 17th century spelling. Black ("Surnames of Scotland", 1946) writes: "Over 250 different spellings of the name have been noted from official sources by Mr. T.C. Kinniburgh, of Folkestone, Kent, who is compiling a history of his family." So little wonder that even the ancestry website has backed the wrong horse.
The place name Conisborough means "king's borough", i.e., a fortified settlement on land belonging to the king.