Scientific site
Despite its lack of natural resource, the island itself is (surprisingly) located miraculously well. It has lots of potential for scientific research. I was thinking astronomy (predictable weather, unobstructed view down to zero-degree elevation) but it could be something else more creative.
It can also be used for testing purposes in advanced applied sciences (weaponry, steam engines * wink * who knows). Being in the middle of almost nowhere, you can test stuff there relatively safe away from prying eyes, both your civilians and your rivals' spies alike.
Auxiliary station, including for emergencies
On the plus side, the island can be supplied with various tools, including repair bases, basic infirmary, and other emergency-related stuffs, just in case your ship needs it and can't afford to wait several weeks of trip to your continental landmass. You will also feel a little bit safer from pirates targeting your merchants because you have this station. If the other superpowers want their ships able to use this auxiliary station, they must be nice to you.
Having such station in the middle of the ocean can have immense usefulness that perhaps can't directly be measured in $$$ for the short term. The usefulness will manifest in other forms, and potentially in the future.
Extending your territorial waters
You can exclusively claim that the seas around the barren island is yours and yours only. 537
If the second superpower want to use the waters: for fishing or other natural resources, voyage path, emergency reasons, whatever, -- now, or in the future, mind you -- they have to ask for permission or setup a treaty or the likes. You've also got yourself an area denial tool or some sort.
Even if your world have no global or multilateral treaties regarding what constitutes a territorial sea at the moment, having the island will benefit you quite a lot in the future when such treaties come into effect.
Justification for exercising power/violence
If this island can be rightfully claimed -- not only by vague constitution wording or claims on paper, but also by actual people living there or at least non-trivially using and maintaining the island -- then after some tens of years it can be used (abused?) as a tool to bully the second superpower. If the second superpower dares to trespass the territorial boundaries or stops there without proper permission, you can claim that the second superpower is violating your territorial integrity and you can justify attacking or interning the trespassing ship with higher confidence. You get to claim to be a defender in front of the international community, not an aggressor, if the situation goes sour.
Forward operating base
In the event that a war actually ensues, having a forward operating base closer to your enemy's home gives you an advantage. When on the offensive, you can resupply your attacks quickly and reliably. When blockading the enemy coasts, you can resupply the blockade safely and easily. When operating a defensive maneuver, you have at least two defense lines: one near that island, and one near your own continental coasts.
It's far better, compared to having no forward operating base at all. What's even worse is having their forward operating base closer to your home. 207