Murray was the son of Thomas Graham Murray and Caroline Jane, daughter of John Tod. His father and grandfather were solicitors, and founding partners of the Edinburgh firm Tods Murray & Jamieson. He was educated at Harrow and Trinity College, Cambridge.
Murray left the government and parliament in February 1905, on being appointed Lord Justice General and Lord President of the Court of Session. He was raised to the peerage as Baron Dunedin, of Stenton in the County of Perth, on 9 March 1905. He held these posts until 1913, when he was appointed a Lord of Appeal in Ordinary. During his tenure as a Law Lord he gave long majority judgments in cases including Metropolitan Water Board v Dick Kerr & Co Ltd concerning frustration and Tredegar v. Harwood concerning a landlord's liability to insure premises, Ellerman Lines Ltd v Murray on employment law and excessive reliance on a preamble or draft international instrument, Sorrel v Smith concerning the tort of conspiracy to interfere with a trade or calling, Leyland Shipping Co Ltd v Norwich Union Fire Insurance Society Ltd on causation in tort, Dunlop Pneumatic Tyre Co Ltd v New Garage & Motor Co Ltd on penalty clauses and Plumb v Cobden Flour Mills Co Ltd on employer's liability. In 1923 he was chairman of the Political Honours Review Committee. He retired as a Law Lord in 1932. Apart from his legal and political career, Lord Dunedin was Sheriff of Perthshire between 1890 and 1891 and Lord Lieutenant of Buteshire between 1901 and 1905. He was appointed a Knight Commander of the Royal Victorian Order in 1908 and a Knight Grand Cross of the Royal Victorian Order in 1922. In 1926 he was further honoured when he was made Viscount Dunedin, of Stenton in the County of Perth.
Family
Lord Dunedin was twice married. He married firstly Mary Clementina, daughter of Admiral Sir William Edmonstone, 4th Baronet, in 1874. They had one son and two daughters. After Mary's death in December 1922 he married secondly Jean Elmslie Henderson Findlay, secretary of the Scottish War Savings Committee in WW1, and daughter of George Findlay, in 1923. They had no children. His only son the Hon. Ronald Thomas Graham Murray was a Major in the Black Watch and fought in the First World War. However, he died married but childless in September 1934, aged 59, predeceasing his father by eight years. Lord Dunedin died in August 1942, aged 92. As he had no surviving male issue both his titles became extinct on his death.