Archibald Cockburn (1738 in Edinburgh, Midlothian – 20 June 1820) was a Scottish judge.
Son of Archibald Cockburn of Cockpen and wife (m. 17 August 1735) Martha Dundas, daughter of Robert Dundas of Arniston (? - 1727) and wife Margaret Sinclair, daughter of Sir Robert Sinclair of Murkle and Stevenston, 3rd Baronet (1643 - 1713), and first wife (m. Holyroodhouse, Edinburgh, Midlothian, 10 September 1663) Lady Helen Lindsay, daughter of John Lindsay (c. 1611 - Tyninghame, East Lothian, 1678), 17th Earl of Crawford, 1st Earl of Lindsay, 10th Lord Lindsay of the Byres, 1st Lord Parbroath and Hereditary Steward of St Andrews, etc., and wife Lady Margaret Hamilton.
A keen Tory, he was Sheriff of Midlothian and Baron of the Court of Exchequer.
He married at Cockpen, Midlothian, on 25 December 1768 Janet Rennie/Rannie, "connected by marriage with Lord Melville" (sister Elizabeth the first wife of Henry Dundas, 1st Viscount Melville), daughter of David Rennie/Rannie of Melville Castle, Edinburgh and had three daughters and six sons: