Thompson, who has been the chair of 3i since 2015 and was Tullow Oil's chairman between 2012 and 2017, will formally start on 5 March 2018, succeeding Jan du Plessis. He was previously appointed a Rio Tinto non-executive director in 2014.
The appointment puts an end to a troublesome process to find du Plessis' replacement, who announced he was leaving in March to take over from Sir Michael Rake as the chairman of BT.
First, former Barclays chief executive John Varley was lined up. Already a member of Rio Tinto's board, Varley was forced to resign this summer after being charged by the Serious Fraud Office over his alleged role in Barclay's controversial Middle Eastern fundraising at the height of the financial crisis.
Conservative party chief executive Sir Mick Davis took Varley's position as frontrunner for the FTSE 100 miner's chair. However, last weekend Rio Tinto caved in to pressure from an investor group representing around 20 per cent of the firm's shareholders. The investors said they would oppose the re-election of a number Rio Tinto's board members at next year's AGM if Davis was picked.
Thompson, who held a number of top roles at Anglo American between 1995 and 2007, said he was "honoured" to accept the role and succeed du Plessis.
He said: "Rio Tinto is in great shape, with a strong management team, world-class assets and a successful strategy."
Du Plessis said he was pleased to be "handing over the baton" to someone he has worked closely with over the last three years.
He added: "The business is in great shape and Rio Tinto has the strongest balance sheet in the sector”.