Okay, here's an answer from 2018, that actually works (I did it on a 2,1 Mac Mini Late 2007). This took me three days of pain and agony and searching to find out how to do it. Hope this helps the three other people doing this within the next ten years.
Ubuntu 18.04.3 (64-bit) installed on a Mac Mini 2007 (2,1) (2.0 GHz model).
Here are the steps I took. Well, the ones that made forward progress, anyway.
Get rEFInd working on your Mac Mini (This might not be strictly necessary, but I used it and it worked and I'm not reinstalling OS X and trying again without it).
Use Rufus to burn the Ubuntu 18.04.3 64-bit ISO (download link) onto a drive, making sure to use the GPT format, not the MBR format, and have the drive formatted to FAT32, not NTFS. I'd assume you could use UUI or Etcher, but I didn't test them.
Download this file and put it in the
/EFI/BOOT/ directory on your newly burned flash drive. (This step came after reading this very helpful article on the Ubuntu forums. I modified what it said to do and it worked perfectly, but do go ahead and check it for yourself if what I'm suggesting doesn't work.) The reason we need this file is that the Mac Mini 2,1 is a 64 bit system that won't do 64 bit EFI boots. So, we need this file to let the Mac Mini do a 32 bit EFI boot of the 64 bit Ubuntu ISO.
Go and plug the flash drive into your Mac Mini, and restart it. When the rEFInd screen comes up, select the fallback bootloader (boot manager? I don't remember) on the flash drive you plugged in. It should have a nice orange icon. Then, when the GRUB screen comes up, select "Install Ubuntu".
Profit or whatever.
I didn't come across any of the issues involving GRUB being broken or not installed or anything, so it was just a clean install and use for me.
A recommendation: As soon as you can, ditch GNOME desktop for Xfce. It allows your new Ubuntu machine to run much faster (i.e. freeze less).
Also, it looks like JWongDev made a pretty good adaptation/improvement on my answer,
You can also install mbpfan, which will sort of make your fans work on the Mac Mini (I still need to figure out how to get it to run at startup).