I think Sector is a pretty solid name for the collection of the 5, though I think Industry might work too. It's not very sci fi sounding, though. Depending on the type of lore you're trying to build, having each of them as a Ministry (or Division) might help establish the world ("Ministry of Agriculture", etc.). Again, though, not especially sf sounding. A quick flip through my thesaurus gave me Stratum (plural would be Strata), which sounds sorta neat. Lamina (Laminae) is a term used in botany/geology used to describe layers, which also sounds pretty sf.

Agriculture is a common enough word that most people will know immediately what that sector does. Cultivation sounds a little more exotic, and is pretty much just as common of a word. Agrology is the science of crop-growing, and, while not especially common of a word, it's meaning can probably be inferred from it's similarities to Agriculture.

Mining is another solid term, because everyone knows what it means. Excavation comes to mind. Something might be able to be done with farm terms like Harvest, but something like Mineral Harvest sounds pretty silly (like a breakfast cereal lol). idk Excavation seems ok

I'm not a huge fan of Industry, and agree with you that Manufacturing seems like a better title, whether or not you call them all Industries. Fabrication might also work.

I like Refining (though Ministry of Purity or something sounds really metal lol). Words with similar meanings include Distill (Distillation), Filter (Filtration), Rarefy (Rarefication isn't a word but hey this is sf do whatever you want. Technically it'd be Rarefying, I think).

I'm not fond of Administration either, but it seems to fit the set of stuff you want to encompass. Obvious alternatives are Government (or Governance), Bureaucratic or Legislation, but I don't think that's what you're looking for. Empire or Regime both sorta fit but not really. idk this one's tough

Something else important I have forgotten?

That depends on how they'll end up affecting the game. There a ton of industries we haven't talked about, but they might not make any sense in the game. What's the purpose of these Industries? How does the Player interact with them? I've been operating under the assumption that the Player just like diverts resources into whatever industries they want and each one has a tech tree or something. Anyways, to my eye, a sort of broader one that's absent is any form of like art or aesthetics, like all this Empire does is work 100% of the time. Which idk maybe they do I haven't seen the lore lol.. And some sort of Commerce/Economics Industry might not be out of place, either. But if you feel like there are enough Industries for the mechanic already, either of those could ostensibly be lumped with one of the existing ones.

One of the benefits of the setting being science fiction, is that if you can write off a bunch of stuff you don't want to implement/don't want to balance and it can still be canon. If someone's like "well why isn't there a Construction industry?" you can just be like I dunno man it's the future, they automated all that, etc. Alternatively, you could show the thing as an industry, but it has one node and it's already been acquired, like "Self-building sites" or "Automated construction". Deus Ex: Human Revolution did that, where there are augmentation trees with one node that's purchased at the beginning of the game because the Devs wanted a canon reason for the Player to start with the ability, like sprinting.

Explain the whole Industry mechanic to me. Not what individual Industries do, but like the system as a whole. Refining affects ship fuel, sure, but can the Player do stuff to make that better? Like level it up with a resource? Or is it like each planet has a main industry and I own X many Refining planets so my Refining level is X strong? Having a firmer grasp on the parts of the mechanic on a whole that you've nailed down will help us understand the situation better, and hopefully come up with an idea or two.

The idea of each planet specializing in manufacturing a specific part sounds ok, though I think the Player would definitely need a way to change (or at least influence) the part being made. You could have it affect combat with things like "this ship gets X bonus as long as you're producing N or more Ys" or "This ship can perform this attack/maneuver only if you have N or more Ys". Maybe even some ships could only be built if you had enough of a thing.

I agree with the Services industry makes sense to contribute towards empire efficiency/happiness. It's up to you how you want happiness to manifest in-game, though. Something like they have in Civilization, where happiness per turn gets put in a bank, and when you have enough, a Golden Age starts, where your empire is more productive/better at combat for 10 turns or so. Or, the happier a planet is, the more malleable to population is to the will of the emperor (the Player has more influence over what happens on the planet). Unhappiness could lead to revolts, mutinies, uncooperative planets, lowered production of stuff, etc.

So a Player can't directly set how much a planet is contributing to an industry, but they can tell the planet what to focus on. Is there any difference between issuing an edict "Do more mining" and going to each planet individually and telling them to focus on Mining? (Are edicts just a way to mass-order the empire?)

A potential idea is to make the AI planet leaders have no predisposition to contribute towards the Service industry? And a reason for the Player to switch a planet from doing something it's good at to Service? idk seems kinda weak, I know. My thoughts are that if a planet is already investing in the thing it's the best in, why would I ever change it?