From  Black’s  “The Surnames of Scotland”¹

The entry for FAIRBAIRN:

Posits that it is a “metathesis for Frebern” and mentions a correspondent of Rev. Robert Wodrow in 1705 spelling the name indifferently as ‘Farbairn’ and ‘Freebairn’.
Stephen Fairburn, burgess of Berwick on Tweed, held the hostelry of the abbot and convent of Arbroath in Dundee a. 1327.
A James Forbrayne, admitted burgess freeman of Glasgow, 1601 may be an erroneous spelling.

Other variants he notes are Faribarne 1536 and ffairbairne 1646.

The entry for FREEBAIRN mentions (amongst others):

Robert FREBORN or FREBERN making a gift of two carucates to the Abby of Neubotle c 1150 and witnessing the grant be Serlo, clericus regis, to the Abbey of kelso.
About 1153 Radulf FREBERN had a grant from Malcolm IV;
Walter FREBERN witnessed a charter by Hugh, cancellarius regis, to the Abbey of Arbroath;
Robert son of Robert FREBORN, witnessed a charter of the third part of Lambertun c 1190-1200;
Robert, son of Robert FREBERN, witnessed a charter by David, son of Tructe in favour of Kelso Abbey c 1200;
John FREEBURN in Polkem, parish of Livingstone, 1702;
Charles FREEBAIN a racksman of Persabus, Islay 1780

NB the more English versions of FAIRBURN and FAIRCHILD are both found amongst descendants of the DNA project’s FAIRBAIRNs, particularly in North America.

PS for some of the assorted transcription errors found, check out the Fairbairn ONS/DNA project diary under Inventive Indexing.