No nation can lay claim to stuff in space. But that does not rule out that companies can, and I guess disputes between them will need to be resolved by international law?
It'll get extremely interesting when a company of country A returns an extremely rare very useful element. Country A will of course lay claim to it, but according to the Outer Space Treaty all space exploration is done 'for the benefit of mankind', so other countries would be able to claim this extremely rare element (or its benefits) must be shared. That'd be hell of an interesting international law case.
It somewhat amounts to "whoever got there first wins".
It seems that the current US position (who will probably be the first to start large scale mining) is that the resource site itself will never be owned but that labor attaches ownership to the ore. As said on the site, this still doesn't deal with when a singular deposit of an element exists and how access would be shared.
Another interesting point that's raised is that if an asteroid is entirely consumed by mining, does that count as 'appropriation'?
Man the next 100 years are gonna be a reaaaallly interesting time for lawyers..