If anyone knows how to get 10.7.3 working, please let me know. The closest I've gotten it to working is the system hanging after displaying the message in verbose mode ' DSMOS has arrived', which someone on said Discord server said sounds like WindowServer failing to start. However, I haven't been able to get 10.7.3 to work on the 2006 Mac mini I picked up for this job. I heard people on a certain Discord server talking about how 10.7.3 was the latest version of Lion that worked on 32-bit Macs (with modifications), unlike what dosdude1 said about how 10.7.2 was the latest working version.
It would be called Mac OS X86, and should be as modernized as possible, with every compatible update installed out of the box, like the latest 32-bit Mac version of iTunes (11.4?) and, if possible, OS X 10.7.3.
I was thinking of creating a modified version of the soon-to-be-abandonware release of OS X, specifically to run on 32-bit Macs that it doesn't support out of the box (maybe even the original Apple TV as well if the drivers are compatible).