I entirely understand what door prize means, but how did this name come about? Is this some kind of slang?
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It's the prize for a contest where the entering is accomplished by walking through the door. In most of these situations, you'd pay a small amount of money to come into a dance or convention, and you'd be given a ticket. At some point the organizers would pull a matching ticket out of a tumbler, and call it out. If no one answered for it, they'd pick a new one. The practice is mentioned as early as 1878.
A "door prize" is a prize given for an event that one gets just by "walking through the door" leading to it. It is a small "loss leader" to get you where the organizers hope to sell you something much more expensive.
As for the early 1950s reference, this practice BECAME POPULAR (in America) in the 1950s, during the growth of the American consumer society. My understanding is that it has a much earlier origin.