It was the female Ferengi that told me that mining claims come from lock boxes in the first place. I clicked on every chat option with her and if I got a mining claim, it is not obvious. I can't see anything that suggests I might already have a mining claim. How do I tell if I got one?
You're trying to do the wrong mission. There are two type of mining. The one concerning the Dominion lockboxes is a much higher yield. You need a mining claim to be able to do that. The other mission anyone can do. You just need a space suit to be able to do "outside" to the asteroid.
Edit: I take back the bit anout anyone. There might be a minimum level required for her Daily mission. But I'm not sure on that.
Be careful DOffing! You could technically DOff your way to the top levels without playing much. Then you'd be a Vice Admiral who doesn't know how to play the game. Talk about getting some rage from players without even chatting!