Do you have a way to mount the old drive on the Mini?
If you need something, get a USB3/SATA docking station, they can be found for less than $25. Go to amazon.com and enter "usb3 sata dock" into the search box, and you will get many hits.
Put the old drive into the dock, connect it to the Mac, turn the dock on, and see what happens.
Does the drive mount on the desktop?
If yes, copy the files you want from it to another drive.
If no, you might try opening Disk Utility, to see if DU recognizes the presence of the drive. Don't do anything else with DU yet.
As the reply above mentions, DiskWarrior can be used to repair directory damage. It's _possible_ that DW might help. Then again, no promises. Keep in mind that DW is intended ONLY to repair a drive's directory, if it can be done. It doesn't do "file recovery".
If DiskWarrior doesn't work, you could try data recovery software such as DataRescue3.
You can try DR3 on the drive "as is". It might help, then again, it might not, particularly if the drive's directory is so badly corrupted that the drive can't be mounted on the desktop.
But that doesn't mean DR3 can't still be used. It's a last-ditch solution, but you might try using Disk Utility to RE-initialize the drive (DO NOT choose to "zero out" the data). With a fresh directory, the drive can now be mounted on the desktop, and DR3 will ignore the "empty" directory and "go right to the platters" to scavenge and rebuild the data (aside: when you re-initialize a drive, all you're really doing is replacing the directory, not the actual data on the platters).
Lastly -- if the drive has an actual "hardware problem", bad motor, broken drive heads, etc., probably the only way to access the data is to send it to a professional data recovery outfit. They may be successful, but be prepared to pay many hundreds of dollars, up into the thousands, to get that data back.
Is it really worth that much to you?
