The government currently controlling the surrounding land? Ethnic tribalism? Genetic essentialism? Protection of the artifact? Exposure/accessibility of the artifact? ‘Feelings’-based reasons like ‘spiritual’ significance or paternalistic rich nations graciously returning things to the poors?
Much to ponder
As you can see, the discussion is already well advanced from the starting point you recommend.
They're more qualified to answer these questions than I am, but my layman's opinion is that a national museum in the country where the artifact originated seems like a reasonable first-blush recipient. There are, doubtless, exceptions.
Wouldn’t this boil down to a might-is-right approach in almost all cases? Don’t really see how a contemporary government annexing the land by force is much different than a foreigner simply taking the artifact. Other than that it might evoke more ethnic/tribalist gripes from people.
But less flippantly: it's no more "might-makes-right" than the original theft/"sale" was, and is arguably less so, given that national museums tend to survive regime changes (usually for nakedly political reasons, such as the incoming regime lashing its legitimacy to the country's heritage).
And on the aesthetic side: all things being equal I, a mostly disinterested Westerner, would rather see artifacts in their regional and cultural contexts than in a dimly lit room in the British Museum.
Clearly the specifics matter, but it gets more interesting when these 2 ‘sides’ have equally tenuous claim to the artifact, as IMO is quite often the case.
I suspect like in most ambiguous situations we’ll get a lot of politics/feelings based decision making. But again I don’t know, which is why I’m interested.
I have been to the British Museum and the lighting was excellent in all rooms.