Where does the term "dropship" come from?

The term "dropship" was used in the December 1983 issue of Isaac Asimov's Science Fiction Magazine in the story "Remembering Siri" by Dan Simmons.

Page 36:

There were only five of us in the dropship. It was always a thrill to me to fall out of high orbit into the atmosphere of a real world.

Used also on pages 38 and 62.

Much earlier, in the July 1960 Amazing Science Fiction Stories in "All the Stars a Stage" by James Blish:

page 84

There was the beginning of a muted stir throughout the Javelin, as battle gear and drop ships began to be readied.

There is a very detailed 1978 description in Analog Science Fiction/science Fact (also in Best Science Fiction of the Year, Volume 7):

Because Celia's father was in the mining business, Lisa Maria batted her eyes at a handsome drop-ship jockey and he took them out to see one of the drop-ships being scrapped in a flickering of cutting torches. These were perhaps the heaviest spaceships ever built by man, rivaling in weight battleships of the Yamato class, but had only one flight in them, down. The jockeys who rode them to Earth called them rafts. They were assembled crudely of massive metal in a low orbit and needed to be able to perform only two functions: to hold together while they burned off their heat shield in the atmosphere and to float when they hit the water. Sometimes they were scrapped at Tongaro and sometimes towed all the way to San Francisco or San Diego or Yokahama.

But the oldest example is not from science fiction. Instead it is an actual military drop ship.

Richard C. Hottelet was a reporter who flew along with the 24 March 1945 Operation Varsity and wrote in a 5 May 1945 article in Collier's Magazine:

hundreds of C-47s flying along in tight formation ...

The three photographers, their cameras clicking away, jostled one another at the waist windows as we swooped around the drop ships.

P-Hour, the drop hour for the paratroops, was 10 a.m.

So the original "drop ships" were C-47s.

The term was popularized in Aliens(1986), where the military shuttle is referred to Cheyenne Dropship. Concept of dropship is very similar to space shuttle, a small craft meant to ferry passengers or cargo around.

What Aliens did, essentially, was to popularize term dropship to specifically refer to military shuttles. Shuttlecraft (another term not from real world, but borrowing term shuttle from the real world) had already been codified by Star Trek to specifically mean small craft meant to travel between mothership and planetary bodies for short distances, as opposed to longer distance made by starships.

Dropships do not exist in real world because the term was specifically coined for scifi. However, there have been concepts of a dropship for Mars travel, where a mothership "drops" shuttles into Martian athmosphere and they travel the rest of the way on their own power. In essence, being dropships, or dropshuttles.

TL;DR

Aliens called military shuttles dropships first (as far as I have been able to find) and everyone else followed the lead. Possible term variants might have existed before, but Aliens is the trope codifier.