Okay, looking at that screenshot, I've had that before and have a suggestion that may hopefully be useful to some people in solving this issue.
In my case, this was a few month ago, but IIRC I was doing the mining when the computer I was on either BSD'ed or there was some kind of power transient that made the computer just turn off. When I started back up and reentered the game, the mining mini game looked just like that, and I of course lost that particular prospecting site with 0 dilithium. (I think I was still below level 50 at that point and couldn't go into the New Romulus adventure zone yet.)
So, what I did and what my recommendation is (if the logging off/quitting and logging back in doesn't work):
1) Quit the program to the desktop if you haven't already done so and restart the launcher. Near the top right of the launcher, there should be something on the bar that says "Options". Select that, and in the dialog box check the "Force Verify" option and then hit okay. I can't remember if you had to restart the launcher or not, but it will do the thing that takes 10-20 minutes to do a file verification of everything in the game. I think in my case some file related to the mining got corrupted because of the timing of when the computer froze or got knocked out, and the force verify found the bad file and replaced it.
2) Once that's done and you can enter the game, go to the "Practice Mining" kiosk inside the Ferengi mining ship to check that it's working correctly. I've taken up the habit of going up to it and checking for a second every time I do mining to make sure, though I haven't had that problem since the one time it occurred.
The relog will fix the problem until you go back to New Romulus and forget to relog and go mining again! Seriously, most of us have had this problem and it's a shame after all these patches they still haven't fixed it!
(Basically, I could hear the sound effects of the triangle turning but the "over sized triangle", which shouldn't be that large wouldn't turn, instead it would just continue to fall in a downwards motion).
In the end I just dropped the "mission". I'm just happy it's not that rich mining one! Haha!