Installation of OS X Snow Leopard on a new hard drive using a disk
This answer assumes you have access to another Mac and a way to connect your hard drive to it.

Take out the drive of your Mac using this iFixit guide.

Connect to the drive and boot into Recovery (CMD + R) on the other Mac. Then, follow these instructions from an Apple help article:


  Install from OS X Recovery
  
  To install OS X on an external drive using OS X Recovery, follow these steps.
  
  
  Start up your Mac. Immediately after you hear the startup sound,
  press and hold the Command+R on your keyboard.
  After you see an Apple logo or globe appear, you can release the
  keys. If you're prompted to select a Wi-Fi network, select a network
  that has a working Internet connection.
  When the Recovery screen appears, make sure you're connected to the
  Internet. If not, select a Wi-Fi network with a working Internet
  connection from the Wi-Fi menu.
  Select the option to Install OS X from the Recovery window.
  Follow the onscreen instructions to complete the installation. When
  you're asked to select a drive for installation, click the button
  "Show All Drives". Be sure you select the external storage device
  you've connected for this purpose, not your computer's built-in
  startup drive.
  
  
  Your Mac might restart one or more times during installation. After installation is finished, your Mac automatically starts up from the external drive.
  
  When you're done using your external disk, select a startup disk from System Preferences to have your Mac to start up from its built-in startup disk or another drive.
  
  Your computer's serial number is sent to Apple as part of this process to help authenticate your request to download and install OS X.


Once the installation of OS X on the Mac mini's disk is complete, insert it back into the Mac mini and follow the iFixit article's instructions in reverse order.