If you no longer have a Recovery HD you can boot from, then you might try Recovery Partition Creator 3.7 – Updated for Mavericks | Musings to create a new one to use. Or if your Mini can use Internet Recovery - Computers that can be upgraded to use OS X Internet Recovery - then you can do this:
Install Mavericks, Lion/Mountain Lion Using Internet Recovery
Be sure you backup your files to an external drive or second internal drive because the following procedure will remove everything from the hard drive.
Boot to the Internet Recovery HD:
Restart the computer and after the chime press and hold down the COMMAND-OPTION- R keys until a globe appears on the screen. Wait patiently - 15-20 minutes - until the Recovery main menu appears.
Partition and Format the hard drive:
- Select Disk Utility from the main menu and click on the Continue button.
- After ** loads select your newly installed hard drive (this is the entry with the mfgr.'s ID and size) from the left side list. Click on the Partition tab in the ** main window.
- Under the Volume Scheme heading set the number of partitions from the drop down menu to one. Click on the Options button, set the partition scheme to GUID then click on the OK button. Set the format type to Mac OS Extended (Journaled.) Click on the Partition button and wait until the process has completed. Quit ** and return to the main menu.
Reinstall Lion/Mountain Lion. Mavericks: Select Reinstall Lion/Mountain Lion, Mavericks and click on the Install button. Be sure to select the correct drive to use if you have more than one.
Note: You will need an active Internet connection. I suggest using Ethernet if possible because it is three times faster than wireless.
This should restore the version of OS X originally pre-installed on the computer.
