The general line on the Bretton Woods system is that it originated as a result of the American desire to protect the massive gold holdings the US accumulated during the 1930s, or that it was supposed to prevent the competitive devaluations of the 1930s, or that it was the product of the war.1
Still, it looks like the basic idea actually emerged from the US going off gold in 1933 and an attempt to figure out what it should do next. In 1934, Harry Dexter White noted you could derive benefits from a fixed exchange rate (easier trade) and from a flexible exchange rate (without having to worry about the exchange rate, you’d be free to use monetary policy to fight off a downturn).
But, White said,
There remains a further possibility in monetary standards. The independence of action with regard to domestic monetary policy may be combined with the maintenance of exchanges that do not fluctuate for long periods of time, perhaps not for several years.
This sounds to me like what became known as the adjustable peg: you derive some of the benefits of fixed exchanges (that’s the peg part) while holding out the possibility of changing the exchange rate in time of need (that’s the adjustable part).
(Having written this post I find James Boughton touched on the same idea on pp. 7-8, here.)
1Of course it is possible there were no competitive devaluations in the 1930s.