At Wikipedia, there is an entry about asteroid mining. Planetary Resources will go after the Near-Earth objects, and take small steps and learn along the way. That seems like a nice business plan (as long as you can wait for any results and profits).
Planetary Resources are far from first to think about Asteroid mining. A snapshot from 1998 at the wayback-machine proves that the BeRKA corporation was in the asteroid mining business 14 years ago.
GDW has thought about asteroid mining even longer than the BeRKA corporation. In the “Consolidated Classic Traveller Reprint Errata” you can find the old “Asteroid Mining flowchart” from JTAS Issue 3.
In the TTA books, there was also some cool images of asteroid mining space ships. Have a look at the PC1 191 Gourmet or the AC3 Stag Beetle. The PC1 191 Gourmet has inspired asteroid mining in MTU. That ship is so much cooler than a Scout Ship with a mining laser.
Except for the flowchart in JTAS #3, a lot have been published about asteroid mining for traveller. To generate belter (asteroid miner) characters for your game, you’ll need Supplement 4. For some fun asteroid adventures, you’ll need Beltstrike! (You can also get Beltstrike for Mongoose Traveller.)
There are also some games about Asteroid Mining. From GDW, there is Belter that is about asteroid mining in the solar system. A cheaper alternative is Asteroids from Radioactive Press, and a fun alternative is Asteroid Racers from Avalon Game company.
You can play the classic Asteroids arcade game at the Atari website. What is your High-Score?