There is no general guidance in the rules
As far as I have seen, there are no official rules for claiming parcels of land. I doubt any rules like this will ever be written, or at least they will be very vague if they are. Historically speaking, claiming land was very dependent on which region you are in, and who you were operating on behalf of.
Some things to keep in mind is that the Forgotten Realms are a medieval-adjacent setting meaning that the rulers of a region typically have total say on land-ownership if they want to (and they often do for fortresses). I would strongly recommend trying to get the go-ahead from the ruler of the region before restoring a fortress to prevent the rule from simply staging their own claim, since theirs would be backed by their governing authority.
This is further reinforced by the appearance of land being granted as an alternative reward in the Dungeon Master's Guide:
A parcel of land is just that, and usually comes with a royal letter affirming that the land has been granted as a reward for some service. Such land usually remains the property of the local ruler or ruling body, but is leased to a character with the understanding that it can be taken away, especially if his or her loyalty is ever called into question.
Since a ruler is the entity mentioned that can grant you land as a reward, it is likely the same place to go to stake your claim.
In your specific case, Axeholm isn't definitively in any ruler's territory. The Sword Mountains are in a state of tribal conflict. However, getting the backing of either, or better yet both the Lords of Waterdeep, and Lord Protector Neverember of Neverwinter (the two closest major settlements) would help keep competing claims off of you, at least from the civilized world (such as Dwarvish heirs). Then you just need to protect the fortress against the orc tribes, and their ilk, in the region.
Example in existing adventures
There is a similar example in existing modules that illustrates how this process might look. In Rise of Tiamat, there is the following excerpt in Chapter 9:
Rian points out that the Zhentarim have a keen interest in Xonthal’s Tower. Ownership of the tower is under discussion by the council, but the tiefling wants to deal directly with the adventurers, who have claim to the tower by right of conquest. The Zhentarim are willing to purchase the adventurers’ interest in the tower for 50,000 gold pieces
As you can see, claims on strategic properties, likely including fortresses, is debatable. However, right of conquest is at least something that exists in Faerun. A suitable, but likely lengthy, argument to either of the powers I mentioned before could land you a deed to the fortress. That being said, once you've restored the fortress, other interested parties will likely make their own claims (whether legitimate or not). Holding such a property is as much a political exercise in the civilized world, as a militaristic one against the tribes.