Recover from backup

I would suggest that you consider using Carbon Copy Cloner for this.

The basic process would be to:

1. Obtain the replacement drive and a either a drive caddy or drive enclosure to temporarily place it in.
2. Connect the caddy or enclosure directly to your mini.
3. Clone the existing drive to the replacement drive.
4. Test the clone by booting up your mini with it connected.
5. If all is well, remove the existing drive and replace it with the clone.
6. You can then optionally place the original drive into the enclosure if you still want to use it as a spare or as a backup.

You'll want to create a recovery partition on the new drive. CarbonCopyCloner, which Tesserax recommends will do this for you as well as copy all your files from the old drive to the new. Like him, I prefer to create the clone before removing the old drive and testing the new one before upgrading the computer.

If you'd rather not pay for CCC (which truly is a fantastic program) you can also download the Mac OS and install it onto the new drive. You can do this while the drive is in the external enclosure or dock. Once that's done boot with the newly made external drive and go thru the setup process. Then let Setup Assistant move your files and applications to the new drive.