From The Surnames of Scotland: Their Origin, Meaning, and History by George Fraser Black (New York: New York Public Library, 1946), page 112:

BUCKNEY. Probably of local origin. There is a Buckney Burn in Stormont district. Patrick Bucknay, heir of Patrick Bucknay, burgess of Linlithgow, 1680 (Inquis., 6257). Patrick Buckney in Linlithgow, 1684 (RPC., 3. ser. x, p. 262).

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There is a Buckny Burn (sic) on this map, near Butterstone, north of Perth—but I don't know if it's the right one.