It's another one of those age old debates because there's no clear answer, and there's no clear answer because the idea kept changing and what we got in the movie was a result of budget cuts and quick merging of separate ideas. This post isn't meant to explain all those ideas, which were explained in different sources, so I won't describe them in detail. But in short, there were suppose to be 2 different locations discovered by the crew of the Nostromo - the derelict ship and a structure that housed the eggs. The structure as designed by H.R Giger was alien-made and almost organic as well, and made out of bones, and resembled a breast. The egg chamber that we see in the movie was suppose to be the inside of the egg silo, but when time and budget dictated that the silo had to go. For budgetary reasons, it was cut and the inside of the silo was moved under the derelict, suggesting the aliens dug their hive like termites
H.R. Giger: Next to the seat of the pilot, there's a hole that leads into depths of the silo (The whole thing has been changed because the exterior views of the silo would have been too expensive!) So the silo was placed under the spacecraft, as if a mini UFO had landed on an anthill, and the ants had eaten their way through the spacecraft, like parasites, in order to use the pilot as a host (Giger's Alien Diaries book)
H.R Giger: We decided that it would be a good idea to have these eggs inside the derelict like termites within the wall of a house (Cinefantastique magazine vol 9, no 1)
H.R Giger: '[W]e had to combine the derelict ship and the hatchery silo. I thought we could place the egg silo under the ship, a bit like termites do.
H.R.Giger sketched how the hive would sit right below the derelict
Dennis Lowe (SFX and Model Cameraman on the original ALIEN): I thought the egg chamber was underneath the derelict too, especially when you see the scale ratio from the model that was built at Bray workshop
And indeed, the CAVE, as Kane describes it in the movie, could not possibly be in the ship. Let's see. The entire derelict in its tallest part is the height of about 11 men
Now, just an unfinished set alone of Jockey's cockpit itself is the height of 8 men, and judging by Giger's concept of the full chamber, and extrapolating from the scale of the set, it would be even more than 11 men in its full height. So that actually corresponds fairly close with the outside of the derelict (the black area at the bottom of the middle photo represents the raised platform that Kane climbs on)
The cave itself, or rather, the part we see, is at least the height of 12 men. So the cave cannot be in the derelict because it's more than the height of the entire derelict, yet it's below the cockpit. So Giger's anthill idea is the only one that possibly works.
But the design of the egg chamber is the same as whatever the interior of the Derelict we had seen, some may say. There's an in-universe and 'Behind The Scenes' answer to that.
The in-universe explanation is that the entire derelict was taken over by the aliens who built the hive/structures around the ship. After all, the aliens build from organic material, like bones, dead bodies and their own secreted resin. It fits their 'architectural' style. And this is actually supported in a way by 'Prometheus', in which we see the ship that is uninfected or rather unaffected by the aliens, and it's neither organic looking, nor made oout of bones like the 'infected' ship from the original movie.
You could even tell the organic material is layered on top of mechanical, which supports the idea. It's even clearer in some other areas:
That explanation was an official one in the Dark Horse Comics lore: The egg silos were indeed Alien Hives, and all that we see inside the derelict and the egg chamber is indeed built by aliens inside their hives.
Insides of the stand alone hives for comparison, from "Aliens: Nightmare Asylum" and "Aliens: Labyrinth"
What adds to the confusion is that Ridley Scott often changes his mind and ideas, and years later decided that the Derelict was actually a bomber carrying bio weapons=eggs, therefore making the cave a part of the ship, even if it didn't really gel like other retrofitted ideas in the prequels. The truth is, there haven't ever been one answer to this, but Giger's explanation is the only one that makes sense and made it into the original EU (extended universe)
