Hi Dirk, some things to try...
Have you done a PRAM reset, CMD+Option+p+r...
http://support.apple.com/kb/ht1379
In fact, do 3 in a row, takes a bit of time.
Intel-based Macs: Resetting the System Management Controller (SMC)...
http://support.apple.com/kb/HT3964
(SMC reset= Shut down, disconnect power cord for 10 seconds, reconnect power cord, wait 5 more seconds, turn computer back on)
but it is slightly different: (PMU REST = Shut computer down, disconnect power cord for 10+ seconds, reconnect power cord WHILE SIMULTANEOOUSLY PRESSING POWER BUTTON, let off power button for a few seconds, then press power button again to restart the mac mini.)
See if it's related to this...
It also seems to be related to later OSX versions, like 10.8.x & 10.9.x
The 2012 macs (and later apparently) (also later OSX versions on earlier Macs can do it also), not waking normally from sleep after hours being in sleep. (noted here ...
http://www.xlr8yourmac.com/archives/feb13/022813.html#2012macswontwake
"Why 2012 iMac/2012 Mac Mini won't Wake After Hours of Sleep (Hibernates/Powers Off)")
The sleepimage file still reappears even if never slept. Delete it (hibernate off, etc) - within minutes its back.
A note on that here - http://www.xlr8yourmac.com/feedback/Chameleon_SSD_Optimizer.html
About iMac sleep...
http://forums.macrumors.com/showthread.php?t=1529750
Try this: "sudo pmset autopoweroff 0" and "sudo pmset standby 0"
http://xlr8yourmac.com/archives/sep13/091313.html#10.8.5SleepEjectTip
Two users reported that the opposite worked...
Try this: "sudo pmset autopoweroff 1" and "sudo pmset standby 1"