Whoever effectively occupies it owns it. Ownership would be significant, in that under the United Nations Convention on the Law of the Sea (UNCLOS), if the new land mass is an 'island' (that being a 'naturally formed area of land, surrounded by water, which is above water at high tide'), sovereignty over the land would give rise to a territorial sea of 12NM around the island and an Exclusive Economic Zone (EEZ) of 200NM. So if it were thought that there was lots of oil in the surrounding area, there'd be a scramble to claim the land. However, UNCLOS makes clear that 'rocks which cannot sustain human habitation or economic life of their own shall have no exclusive economic zone or continental shelf'. Same goes for 'low-tide elevations', or bits of land that are not exposed at high tide. So if this new land mass was not capable of sustaining human habitation or economic life, claimants would only be entitled to a 12NM territorial sea. This could still be highly desirable from a strategic standpoint, but not so much from an economic one.
That largely assumes that states are willing to abide by UNCLOS and not stretch the facts to suit their agendas. If the land mass were simply a 'rock', it's likely that there'd be a rush to claim it anyway, and whoever succeeded would seek to establish an EEZ. By this stage the issue is more diplomatic than legal. A solid legal claim will be very helpful in diplomatic negotiations, but it's not guaranteed that a state's claim will be accepted just because it has a good legal claim. If you saw a new land mass in the South China Sea, it is highly likely that China would occupy it and subsume it into its 'nine-dash-line' claim, even though there may not be a solid basis for that in international law. Other states could complain all they like, but it wouldn't change the fact that China would have the upper hand in any negotiations, and would likely be able to either extract concessions or continue on without the existence of any sort of international consensus. In fact, that's precisely what it does already.
Edit: I don't know my units very well.

source: If a new significant land mass appeared in international waters due to volcanic activity, how would it be decided who owned it? : AskReddit