What version of OS X are you running? Use specific version numbers (like 10.4.6, for example).
Is your Mac mini going to sleep when this happens? Remember that unless you tell it not to, the Mac will disable ethernet when it goes into sleep mode.
The other option would be to zap the PRAM by holding down Cmd-Option-P-R at the same time right after the Mac startup chime. Hold it down for three or four chimes and then let go. (BTW, the "Cmd" key is the one with the apple and propeller icon on it.)
• Apple iMac G5 17" (2 GHz G5) - Mac OS X 10.5.8/Ubuntu 10.04
• Asus Eee PC 901 (1.6 GHz Atom N270) - Fedora 13
• Apple Macintosh Quadra 650 (33 MHz MC68040) - Mac OS 8.1
• "JHVH-1" (2 GHz AMD Athlon XP 2400+) - Slackware 13.1