The term American Dream hasn't always been used, and when it was first used it wasn't intended to be used the way most think of it today.
The Library of Congress tell us that James Truslow Adams, in his book The Epic of America, which was written in 1931, stated that the American dream is "that dream of a land in which life should be better and richer and fuller for everyone, with opportunity for each according to ability or achievement. It is a difficult dream for the European upper classes to interpret adequately, and too many of us ourselves have grown weary and mistrustful of it. It is not a dream of motor cars and high wages merely, but a dream of social order in which each man and each woman shall be able to attain to the fullest stature of which they are innately capable, and be recognized by others for what they are, regardless of the fortuitous circumstances of birth or position." (p.214-215)
Here's the history of the usage of the word. As you can see when it really took off is when American Dream was picked up by National Home Builders Association as a way to encourage more home ownership.
I researched this in early 1990s and right after I'd discovered this history of the usage, the fact that Truslow had first used it in his book The Epic of America, I came across a copy for just a couple of buck in a used book store.