There were a lot of reports of similar problems when the new shape Mac mini came out. As it did not affect me personally I don't recall seeing a definitive fix.
There is the Doctor, Doctor approach...
Patient - Doctor, Doctor it hurts when I lift my arm
Doctor - Well don't do that then
In other words use a modern digital monitor with DVI or better.
However other things you could try doing are check to see if your Apple Mini Displayport to VGA adaptor has had its firmware updated as per http://support.apple.com/kb/HT3492 and you could try using Screen Sharing to remotely control the Mac mini and then be able to select various different resolutions in Displays until you hopefully find one that matches your VGA monitor and works (it might be defaulting to a setting that will not sync properly).
I plugged the VGA adapter into my MacBook Air Thunderbolt port and got a display. Then I plugged it back into the MiniMac and the display is working! Only thing I can think of is that the adapter wasn't fully engaged into the Mini Mac port the first time I tried it. Thanks for your quick replay.
A further update to our monitor problem:
After posting my last message I noted that when the display/computer goes into screen saver or sleep mode, the screen was blank with the "Check Cable" message scrolling across the screen. Unplugging the Thuderbolt connector at the computer and plugging it in again was the only way to restor the images from the computer.
This morning I replaced the original cable (and VGA connectors) with a cable that was twice the diameter of the original cable. The monitor has worked flawlessly since then at re-start from the sleep state. I conclude that either the replacement cable, connectors or both do a better job conveying the signals between the computer and monitor. It could be that there is some high speed signal the monitor looks for to initiate re-start that is distorted by the original cable.
With either cable the images and text are crisp with no distortion. I see no need to upgrade to a DVI cable and connection between this monitor and the Mac Mini now that the monitor re-start function is working.
I am glad you have found a solution. The problem of the signal being lost when sleeping and not being restored when the computer wakes is one a lot of people have had with that VGA adaptor.
The signal should be lost when the Mac sleeps the display (even if the CPU is still active) as the Mac turns off the video port. The monitor will then not see a signal and will as you report show that, often the monitor will then go to sleep which is usually what is desired. Obviously when the Mac is woken the signal should be restored and the monitor should see that and wake itself.
Earlier reports from others suggested using the DVI adaptor instead of the VGA adaptor was a reliable solution.