Old Mac Pro starts randomly hanging - how do I figure out what's wrong?
Is Lion Server a fresh install, or is it updated from a prior OS? My suspicion is that you're likely suffering from a hardware issue (the only times I've experienced that sort of problem, HW was the culprit), but your first step is to get it running as vanilla as possible and see if you're still having the issue. 

Best bet is borrow a monitor, run diagnostics on the boot drive (make sure the drive isn't dying and the culprit), wipe the drive and install the OS fresh. Confirm you don't have any weird third party hardware connected and see if you're still having the issue. RAM is notoriously finicky, so I'd also look into that (although I admit I've never heard of janky RAM causing this sort of problem) - if the RAM has been upgraded at some point, try just running the original RAM that came with the machine when new. 

If you can eliminate software being an issue, then at that point you'll know a hardware tech (Genius Bar or otherwise) would need to root out the hardware cause.