I don't see that on the MINT site, but found it in Softpedia.
MATE is a GUI alternative?
It's on the Mint Download page:
https://www.linuxmint.com/download.php"An edition featuring the MATE desktop"
The biggest issue here for a lInux system is likely the specifications in question.
You haven't actually detailed the specs you're aware of, but even assuming maxed out specifications for the model things might be problematic:
600Mhz Pentium III
320MB of RAM (maximum 256MB Slot +64MB soldered)
ATI Rage Mobility 4MB. This is based on the ATI Rage Pro.
These specifications present a challenge for finding an OS to run, and I'd argue likely eliminate modern distributions like Ubuntu or Mint, instead probably requiring one look into the distributions built for older systems or systems with limited capabilities, such as LUbuntu, VectorLinux, etc.
Just by way of example, I have a Thinkpad T41p which is a more modern system and it was unable to run Linux Mint without the cinnamon desktop constantly crashing and major graphical problems; MATE could run fine, but that system had a much more recent Radeon 7500 M chipset, much more memory, a faster processor, etc.
back to the booting issue- it is not possible to boot the Thinkpad X20 from USB at all, so unfortunately flash drives, external CD/DVD drives, etc are all out. There are external CD/DVD drives for it which use a proprietary connector interface that can be used, but I doubt those can be found cheaply and I suspect you'd rather not throw money down a hole on that sort of thing.
This leaves Patio's suggestion- install Windows onto the HDD from another system, then replace it into the computer. I'd recommend copying the contents of the Windows CD to the HDD after doing the initial install on another system, as well, in case Windows requests driver files during boot-up when you replace the drive in the Thinkpad.
Once Windows is able to boot, you'd be able to use External CD/DVD USB drives as needed further, to transfer additional software for example.
Another option could be to install MS-DOS (Assuming it has a Floppy drive) to at least have it have an OS.
My thinking when I originally suggested selling it is because people "collect" models like the Thinkpad; even without a CD/DVD Drive if it is in fairly good shape I think it would be worth selling it that way, as the alternative would be to throw your money down a hole on a drive to get it booting and it's not a particularly capable system either, so you'd have to decide if it's worth it. Chances are the sort of fella who would buy it off eBay/craigslst/etc. may already have the necessary parts to get it booting.