OK, I may have a solution. It worked only in Developer Previews on Core Duos, but since it just tricks the installer and you now have a Core2Duo it should work...
Use the same method as there. http://forums.macrumors.com/showthread.php?t=1107457
You should do this manipulation on the installation disk image of Lion, just check how to extract it from the installer.
Use this to extract the disk image: http://www.youtube.com/watch?v=kPXe0HeD-cY&feature
Copy InstallESD.dmg to your desktop mount it and do the workaround I just mentioned (delete the .plist file, see the first link) on the image... Then, just burn the image to a DVD or to a external drive or USB-Stick (follow the instructions in the video I just mentioned)
Finally, on the DVD or drive you just created, click on the Lion Installer once it is mounted in the Finder and try to install it. This should work.
Try this workaround instead of the one I mentioned before (the one with the hard drive cloning) and backup your data instead it doesn't work, so you can still go back to Snow Leopard.
Can you please notify me if it works, because my sister has an old mini too, so if it works I may take it apart and put a Core2Duo in it and to the same trick too. I'm waiting for feedback, since i don't want to buy a processor if this doesn't work...