According to the Wikipedia, that exact phrase never appeared in the original Star Trek, though a similar one did: "Scotty, beam us up."
Which is too bad, because I was all ready to write a brief note on why that phrase sticks in the mind in the way Picard's "Make it so, Number One" really doesn't. Here's what I was going to say.
For one thing, it's short, only four words. Picard's phrase is short as well. But it's not particularly suggestive of anything while "beam me up" evokes perhaps the most interesting bit of futuristic technology in the Star Trek world, the transporter. On the one hand, it's miraculous; but then so are replicators, warp drive, and the universal translator. But it's visually more interesting than any of them.
And so it becomes emblematic of the show and of the Star Trek universe. "Make it so" just doesn't have that power.
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FWIW, Google's Ngram viewer first picks up "beam me up Scotty" (note that it's case sensitive) in 1i82.
