Lion refuses to download from App store stating " unable to install on this machine"
Apple determines Lion compatibility based upon the original shipping configuration of your machine. Since there was never an "official" upgrade offered from Core Duo to Core 2 Duo, according to Apple your machine is not supported.

But...

There are workarounds out there, if you are willing to get a little creative.

The simplest method is to install Lion onto a drive using a supported machine.  Once Lion is installed, it will boot and run just fine on your upgraded Mac Mini.