What they were trying for, were more NASA mission control type commands really. Almost truckers in space, on a CB radio.
When something's ready, it's "Good to go" for instance.
"Let's go" probably got repeated a few times. Not really catchphrase material though.
I don't think you can genuinely try to manufacture that kind of thing. Often what gets remembered, wasn't said that often... or in the exact way it ends up being recited anyway.
"Beam me Scotty" breaks down as three words rearranged or extrapolated from what actually got said during TOS.
"Engage" was coined during their unsold Pilot and virtually disappeared until TNG came along.
It became "Mr. Sulu. Ahead, Warp factor [whatever]".