I guess the installer checks the model number, the ROM or something else that identifies the model of the Mac.... I guess you can make a workaround by modifying .plist files in the installer. Probably it doesn't check your hardware, but only the model number, so it probably thinks you still have the standard Core Duo mini. As long as no workaround is found you may try to install Lion on an external drive with a supported Mac (Core iX, Xeon, Core2Duo) and then clone the content of this drive to your minis hard drive with a utility like CarbonCopyCloner or SuperDuper.
Clearly:
1. on a supported Mac, create a bootable Lion installer, boot on it and install the system on an external drive
2. once it's installed on the drive, plug the drive to your mini and hold alt while booting, if the external drive is listed, select it and boot on it... If this works, you're lucky, if not, you may wait for a workaround. :/
3. once you booted install CarbonCopyCloner on the external drive with Lion in it and clone it to the startup-drive of your choice (ex. internal drive)
4. once done, reboot on the newly cloned Lion drive (you don't need the external one anymore)
NOTICE: WHEN YOU CLONE YOUR LION DRIVE TO YOUR HARDDRIVE, EVERYTHING WILL BE DELETED. SO BACKUP YOUR DATA ON ANOTHER DRIVE AND COPY IT BACK MANUALLY TO YOUR NEW LION DRIVE!!!!!