I have this mechanic: there are planets and the player assigns some of the planets as "administrative centers" (or we could call it other way, like cultural center, trade center, but I like the administration center the best so far). Then planets that are near the administrative center join that "administrative zone" and share some resources, traits, etc.

Now, how to call this administrative zone (which consists of one "central" planet and several neighbouring ones)?

The most SF/space emperish sounding one would be "sector", but to me sector gives an impression of a "rectangular thing on the map" and the zones would of quite unpredictable and frequently odd shaped ("quadrant" would be even worse here, it's not quad at all :)).

First thing that comes to mind is 'district' (but you've already thought of that, as I now notice it's in the thread title).

You might also call them 'Cantons', though I think that'd only work well if each canton is explicitly named - probably taking the name of the administrative center planet.

e.g. if Trantor is the name of one of your administrative centers, then the canton it controls would be the Trantor Canton / Trantor District / Trantor Region.

Here's a list of english words meaning 'administrative zone'. Oblast sounds cool. More 'fringey' administrative zones could be called Provinces.

if Trantor is the name of one of your administrative centers, then the canton it controls would be the Trantor Canton / Trantor District / Trantor Region.

O! Great idea, I was worried how to name these adm.zones, "PLANETNAME adm.zone" sounds logical and thematic.

I have digged through the list you linked I think these might fit:

Department (bureaucratic sound to it)
District (not sure, isn't it too much like a part of a city?)
Prefecture (Holy Roman Empire in space :D)
Province (sounds monarchish)

Which is the best and do you have other proposals?

District or zone sound more like a region based union, whereas your presentation of the administration union sounds more like a political/organisation union. Therefor I would prefer Province or Prefecture. Empire or federation doens't sound modern too, yet they are frequently used in space sci-fi.

About the game (maybe this would help):

The game is called "Pocket Space Empire", you are the Emperor (you grant audiences, appoint governors, etc). The feel I aim for is partially "old bureaucratic empire with high corruption" and maybe *slightly* feudal (but really slightly). Definitely it's not high tech, there are plenty of planets with poor colonists that live in poorer conditions than we live now, there is technology, but it's expensive and not everywhere common people have access to it. So, the feel completely unlike Star Trek for example.

if Trantor is the name of one of your administrative centers, then the canton it controls would be the Trantor Canton / Trantor District / Trantor Region.

O! Great idea, I was worried how to name these adm.zones, "PLANETNAME adm.zone" sounds logical and thematic.

I have digged through the list you linked I think these might fit:

Department (bureaucratic sound to it)
District (not sure, isn't it too much like a part of a city?)
Prefecture (Holy Roman Empire in space )
Province (sounds monarchish)

Which is the best and do you have other proposals?

I like 'district' and 'prefecture'.

To me, 'Department' sounds like a physical location ("go down to the marketing department"), or a conceptual group ("marketting") designated based on the tasks or roles. Since every location in an empire has arbitrary differences and similarities, I don't think "department" would work too well.

You could also use different zone names based on how close it is to the empire's center: 'District' (nearest 20%) -> 'Prefecture' (60%) -> Province (farthest fringe 20%).