There a lot of rare earth metals, and radioactive whose demands are increasing and the supplies are shrinking on earth. There lots of reason to go space mining, on the moon or on near earth astroids. And the Moon has an abidance of an element that is very useful, but is nearly none existent on earth, H3.
Rare Earth metals are found in abudant supplies on Earth(You find them everywhere). There just ain't much decent profitable deposits. The recent shortage is only caused by a spiking demand and an inconsitent supply(ak, 80% of the supply is delivered from a single mine in China). Once the new mines in the USA and Australia open, prices will stabilize back to their normal levels(Which are less than one tenth of the current levels). Besides, even if there were solid deposits of Rare Earth's on the moon(as in 100% pure), the prices still need to double to cover for the launch costs alone.
Asteroids are more logical than the moon, because of the smaller launch costs, and the much higher concentration and accesibility of resources. An asteroid mine is temporary however. So a single mine wouldn't be multinational, but the whole project would be. However, in order to cut on launch costs, these mines will all be very long term projects(Ie, long preparation, short operation of the mine), and might be better attempted in a group.
Actually, it's called He3. Which is a difference, since H3 would be an isotope of hydrogen(I believe tritium, which is not found on the moon, but can be found and produced on Earth. And is used for first gen fusion), while He3 is an isotope of helium (Which is found on the moon). However, I need to add that He-3 is completely useless without second generation fusion, and can be found in such amounts on the moon that there's no reason to go for a riskier single nation moonbase rather than a combined project.
Also, there's quite a lot of cooperation between space organisations already. The ESA uses Russian rockets for Medium-small loads, and in exchange the Russians get to use some of out launch sites. There are also contacts with China, and with NASA.