It would probably be based on a homesteading system. In the case of a planet, you'd get what you changed. In the case of an asteroid, if you moved it into a different orbit and set up facilities on it that's probably enough to count as a homestead.
Over 200 asteroids are known to be larger than 100 km,[49] and a survey in the infrared wavelengths has shown that the asteroid belt has 0.7–1.7 million asteroids with a diameter of 1 km or more.[1]
The volume of a sphere 1 kilometre in radius 4.19km³. If you can build a mining operation on one of those I don't see any problem with granting you exclusive access. We already do something similar on Earth.
