Hey TechUnwizard,
I'm sorry you're running into this hard drive issue. You might want to check your hard drive cable first, just to see if by some crazy chance it isn't properly seated or sometimes just reseating (unplugging it and plugging it back in) can help. Also, do you have another laptop that you could try putting it in? Sometimes the hard drive cables go bad and putting it in another laptop will help you test that.
Yet another test would be to see if it is shown in the BIOS.
1. Press F10 at power on.
2. Arrow over to System Configuration
3. Arrow down to Boot Options and press Enter
4. Arrow down to Boot Order and press Enter
5. See if the Internal Hard Drive is listed there
If you try that and it is still isn't recognized, you can replace it with any SATA 2.5" laptop hard drive The capacity is up to you (and your budget, although, larger capacity hard drives are a lot less expensive than they used to be).
I would say your computer is probably SATA II (3.0Gb/s) but SATA III (6.0Gb/s) is more popular and it is backwards compatible. So you could use SATA III, you just wouldn't be able to optimize the speed of the hard drive with it in your mini. As for the RPMs, 5400 or 7200 is probably what your current hard drive is, so I would say get at least 5400, 7200 or higher. If you have any questions about those specs, let me know your product number. It will be on a label on the bottom. Hope this helps! :)