The Answer:

Probably the most famous modern use of the phrase, responsible for much of its popularity, was in 1941's The Maltese Falcon. In the film, Sam Spade (played by Humphrey Bogart) refers to "the stuff that dreams are made of."

Spade was paraphrasing a much older expression. As tends to be the case an awful lot of the time, this dates back to Shakespeare. In Act IV of The Tempest, Prospero says "We are such stuff / As dreams are made on, and our little life / Is rounded with a sleep."

—The Editors