The 48-year-old Scot, winner of 13 grands prix and now a TV commentator, succeeds ex-rally driver Paddy Hopkirk.
The BRDC owns Silverstone, home of the British Grand Prix, and regards itself as guardian of British motorsport.
BRDC chairman John Grant said: “In David, we have a president who is not only a greatly talented racing driver but also a successful businessman with an enviable network of contacts in the motorsport world which, together with his experience, should be of great benefit to the club.”
At the same time as Coulthard was elected president, Indycar legend Dario Franchitti was elected a vice-president.
