From "might makes right" is Tongan because the Tongans were able to take it. From the perspective of homesteading, owning the fruit of your labor, etc I think you can argue since the Minervans both made the island, homesteaded, "discovered" (didn't exist until they made it), and claimed and were actively using it they have senior property rights (you can argue any of these individually but in summation it's quite weighty over any Tongan claim).

Of course might make right ultimately trumps everything else, it is just interesting that you so often hear that if libertarians want to escape society they shouldn't use force to make others follow their ideals, they should just go off into the woods or their own island or some such. But then when they actually make their own island, actually "society" decides they will just take their shit under the auspices of a military force that will kill them if they defend themselves (although the only homicide on Minerva was one Tongan killing another Tongan).

Right now Fiji and Tonga are fighting over it and in reality neither one actually gives much a shit about the actual property rights to hold the island and as a fiji/tongan dispute suddenly the Navy is not so interested anymore. The Tongan claims were initiated after a conference with Polynesian countries and Australia where the goal was not to preserve some Tongan fishing but to smack down the libertarians using it -- Polynesian claims were never an actual reason for the invasion, only kicking the libertarians off the island.