According to http://www.everymac.com the mid-2007 Mac Mini has a Core 2 Duo processor. Click the apple in the left corner of the menu bar, About This Mac and see what processor you have. If you have the Core 2 Duo you can upgrade to Lion, 10.7.5, if you have at least 2 GB ram, though it runs better with 4 GB, and 7 GB free space on the hard drive.
Otherwise you can only go to Snow Leopard, 10.6.8. In any case you will need to purchase Snow Leopard from the Apple on-line store for $20, install, on-line update to 10.6.8, and then if you can go to Lion call Apple at 1-800-MY-APPLE and purchase a code to download Lion from the App Store.