How to upgrade hard drive in Mac Mini
Solution 1:
You could use Apple's OS X Recovery Disk Assistant to create a bootable recovery disk on a USB thumb drive. Since you've backed up to an external disk using Time Machine you could replace the hard drive, boot to the recovery disk on your thumb drive, use the recovery disk to format the hard drive and then reinstall 10.8/Mountain Lion. Once Mountain Lion is installed you will walk through the same setup procedure that occurred when you first bought your Mac. One of the first questions in the setup process is if you would like to restore information from Time Machine, which allows you to connect your Time Machine disk and restore your backup.
Solution 2:
I recently did this very thing. I backed up my OS and files to an external drive (not a thumb drive), installed the new drive with new OS, and restored the Time Machine backup to the new drive. The whole process took a couple of hours. The best online video for doing this operation IMO is here: http://eshop.macsales.com/installvideos/mac_mini2011_hd/
Replacing the hard drive will not void your AppleCare warranty. OWC addresses this issue on their blog: http://blog.macsales.com/18244-owc-diys-wont-void-your-macs-warranty
NOTE: I have done the restore process in #1 using both Time Machine and Carbon Copy Cloner. Each is different, but relatively simple.