Maybe the VGA/DVI cable isn't plugged in properly. Try unplugging both ends and blowing in the ports to remove any dust and then plug them back in securely. It could also be the adapter you're using, if you've got a spare one, try that one.
In the worse case, it might be an issue with the Mac-Mini itself - maybe the port it was shipped with is damaged somehow. If this is the case, I'd go to my nearest Apple Store to get it fixed.
Hope this helps,
In Energy Saver Prefs, make sure Put Hard Disk To Sleep When Possible is (Un) checked.
Good luck, Tom