Stephenson was the son of Henry Frederick Stephenson MP, and Lady Mary Keppel. His eldest brother, Sir Augustus Keppel Stephenson, was a Treasury Solicitor, and the second person to hold the office of Director of Public Prosecutions in England and Wales. On 18 December 1855 Stephenson joined the Royal Navy, becoming a Naval Cadet in HMS St Jean d'Acre, commanded by his uncle Henry Keppel, and serving in the Black Sea during the Crimean War. From September 1856 to April 1857 Stephenson served under Keppel as a cadet in HMS Raleigh, serving in the East Indies and China during the Second Anglo-Chinese War, until his ship wrecked near Macau when it struck an uncharted rock. All the crew were saved. In June 1857 he served as a Midshipman in HMS Pearl, serving with Pearls Naval Brigade during the Indian Mutiny of 1857, during which he was Mentioned in Despatches three times. In June 1861 he was promoted lieutenant in HMS Emerald, serving in the Channel Squadron. On 30 March 1866 Stephenson was the lieutenant-in-command of HMS Heron, serving in North America and the West Indies, and becoming the commanding officer of a gun-boat on the Canadian lakes during the Fenian raids of 1866. From 18 January 1867 to 26 April 1868 he served as a lieutenant in HMS Rodney, commanded by Algernon C. F. Heneage, the flagship of Vice-Admiral Henry Keppel, serving in China. Following the death of Commander John T. Swann, Keppel promoted Stephenson to commander on 26 April 1868; the promotion was confirmed by the Admiralty on 7 July 1868. From September 1868 to August 1871 he served in HMS Rattler and HMS Iron Duke, serving in the Far East, and later in HMS Caledonia in the Mediterranean During this period he also served in the Royal YachtVictoria and Albert. Promoted to captain on 6 January 1875, from 15 April 1875 he commanded HMS Discovery for the British Arctic Expedition of 1875–6, led by George Strong Nares in HMS Alert, as a result he was appointed a Companion of the Order of the Bath on 9 December 1876. He was appointed Equerry-in-waiting to the Prince of Wales on 5 July 1878 he held this post from time to time until 4 April 1893, when he was appointed an Extra Equerry. On 15 September 1880 he became captain of HMS Carysfort. He participated in the recapture of Ismaïlia, and was awarded the 3rd Class Order of Osmanieh by the Khedive of Egypt in 1883. He was appointed Aide-de-camp to the Queen on 1 January 1888. He was additionally appointed CB in the military division on 23 May 1889.
He married the Hon. Charlotte Elizabeth Eleanor Fraser on 5 December 1903. She died in 1923 and Stephenson died at home in London on 16 December 1919 aged 77.