Laser mining the rocks you've mined will refresh after 2 hours (provided you leave the instance). Finding the same rock a second time generally requires a good reference point and a lot of practice, though.
The other types of mining do also refresh but it takes a while.
On the other hand, these ring systems are huge - you're not going to clean one out on your own. It's generally only noticeable if you're deliberately mining in exactly the same spot (e.g. a RES, or dead centre of a hotspot) over and over.