Hi, you can upgrade your Mac Mini to El Capitan (10.11) which is the end of the road I'm afraid. You will at least be able to run a more recent version of Safari but I'm not sure if your Mac Mini will receive Security Updates. It's possible but I can't verify this. Maybe someone else will know this info.
Try the link below when you are ready to upgrade, let us know if it doesn't work. I tested it out & was taken to the download in the App Store. Once you download it then El Capitan will be available in the future if you open the App Store & click on "Purchased". Another thing you could do is when the download is completed & the installer launches automatically, you could quit it then open your Applications folder & make a copy of the installer to another location. Then launch the Installer & click continue and follow the onscreen instructions. I couldn't find a link for Mac OS X Yosemite.
How to download OS X El Capitan
It's extremely important to make a backup, whether you use Time Machine or use a 3rd party app like Super Duper etc, before you start the upgrade.
Anyway good luck & let us know how you get on.