You COULD do this with a Snow Leopard commercial installer. The problem is that your 2010 mini originally shipped with version 10.6.3 -- but was a special build. The commercial installer with the same 10.6.3 version won't boot or install on your mini. BUT, you can do this, with the commercial 10.6.3 installer, if you have another Mac that is slightly older than your 2010 mini. For example, a 2009 mini. Connect the two Macs with a FireWire cable. Boot your 2010 mini to Target Disk Mode. Then, with a Snow Leopard installer DVD in the older Mac, boot the older Mac to the Snow Leopard installer. Install Snow Leopard on the drive in the older Mac. Let the installer complete, set a user up, and then run Software Update, so it will then be updated to 10.6.8. That updated system will now boot the 2010 successfully in 10.6.8 (or it should)
The 10.6.7 version that startergo suggested might work, but would be a copy of the special build of Snow Leopard used on a different Mac, and may not work in every case on another Mac. Apple never sold a full installer later than version 10.6.3, so unknown if that 10.6.7 will work correctly on a Mac that is not part of that specific install set. So, might be worth a try. I wish you luck with that version.