so in my case, I was upgrading the original 250G SSD to a new 512G SSD. I installed the new drive in the MacMiniServer first, then used a SATA-to-USB cable to connect the original drive externally. Next you need to boot off the old drive, but - it has to be from the Recovery partition of the old drive, since you are going to be cloning the active boot partition and you'll need those files left unlocked to be copied.
The reason for my comment is that I tried using CarbonCopyCloner first - everything went smoothly but when I booted up on the new drive, the OS would reset itself every 20 seconds or so. Apps would not even launch before the reset.
Not sure why I tried CCC first - maybe because I had a copy...
Anyway, the trick is to connect up the drives as mentioned above, then hold down the Option key when you power on the MacMini. This will bring up a grey screen and allow you to select which partition to boot from. Choose the externally connected Recovery Partition. Once booted, you can select Disc Utiilties. Tab across to the Restore Tab, select your Source Drive (the external USB connected old drive) and the Target Drive (the new SSD installed in the Mac Mini) and fire away. Takes a few hours, but it seems to copy something across that CCC missed.
I've not tried Time Machine for restoring or migrating a system before - I was under the impression it just restored whatever files you had targetted for it to backup (which may not be everything you need). For my case, I wanted the 250G partition to be cloned and migrated onto a new 500G partition to give me more space.
Hope that helps - Merry Christmas to all ;-)
Sean