In no particular order...
Export your boot disk to an external disk using Disk Utility booted from an installation DVD, and test-boot from that. (Hold the option key during a power-up to get the chooser, and select the external disk.) That's (usually) a viable recovery path, and worth testing.
Your boot drive will likely run from a similar-enough Mac Mini or Mac Mini Server, but there are a variety of Mac Mini Server boxes that are too new to boot and run OS X Server 10.6.8. Test any replacement.
One of the common approaches for a reinstallation-style upgrade (if this hasn't already tried) is a clean install followed by a migration in from your existing disk; either directly (via Target Disk Mode) or from an external copy created via Disk Utility (as mentioned earlier).
If your disk isn't upgrading successfully, then there is clearly already something a little wonky. Or corrupt. And if Apple couldn't sort it out short of an installation, you're potentially operating on borrowed time; these wonkies can have a wonderful habit of (re)appearing or mutating at the most inopportune moments, too.
One alternative option involves cloud hosting or such, depending on exactly what you're using this server for; outsourcing the server and the hardware. (This assumes various details of course, and might or might not be feasible in your particular case.) Or yes, get the budget in place for a replacement (new or used) Mac Mini Server, and test with the migration path "mentioned" earlier.
