Henry Cockburn CB (2 March 1859 – 1927) was a British diplomat.
Cockburn was born in Calcutta in 1859. He was a son of Francis Jeffrey Cockburn (Edinburgh, Midlothian, 8 January 1825 – Brentford, London, 10 July 1893), a Judge in India and with the Bengal Civil Service, and wife (Calcutta or Westbury, Tasmania, 25 January 1855) Elizabeth Anne (Eliza Ann) Pitcairn (Hobart, Tasmania, 23 September 1831, bap. Hobart, Tasmania, 7 November 1831 – Wycombe, Oxfordshire, 1923). His paternal grandparents were Henry Cockburn, Lord Cockburn, and his wife Elizabeth Macdowall, while his maternal grandparents were Robert Pitcairn (Edinburgh, Midlothian, 17 July 1802 – Hobart, Tasmania, 1868) (son of David Pitcairn and Mary Henderson) and his wife (m. Hobart, Tasmania, 30 September 1830) Dorothy/Dorothea Jessy Dumas.
Claud Cockburn, the journalist, was his son and the journalists Alexander Cockburn, Andrew Cockburn and Patrick Cockburn are his grandsons.
Cockburn served in China for 25 years from 1880, as British Consul-General to Peking and Vice-Consul in Chungking, China. In late 1901 Cockburn was appointed to assist Sir James Lyle Mackay, who had been appointed His Majesty´s Special Commissioner to conduct negotiations with representatives of China, The negotiations resulted in the Sino-British "Mackay Treaty," which anticipated the abolition of extraterritoriality in China.