Linux

Rebecca Black OS.
It is the only Linux distro to date built around Weston, using Wayland's full capability:

It doesn't include any Rebecca Black theming or is related to her in any way.
It's just called that cause the dev is a fan of hers.

DSLinux

Hannah Monata Linux and Red Star from North Korea.

The first one that came to mind was fli4l (Floppy ISDN for Linux). Originally a distro of German origin that fit on a single floppy disk to turn a 386 or 486 PC into a router for ISDN connections. Last I looked it's still actively worked on.

There are probably tons of more obsuce ones. But this is one I actually used.

Check out the random button on Distrowatch (distrowatch.com/random.php) - it's like a Linux lottery, but you always win something weird!

Smoothwall. I used to run it a lot back in the early 2000s for personal use and even helped set up a couple small businesses with it but I don't hear of anyone else using it these days, people seem to love openwrt and pfsense more.

It was great for just taking any old x86 machine and making a powerful, fully featured firewall/router out of it, including a VPN server, all through a web interface. Nowadays that's boring shit but in 2002 it was pretty cool.

elive

you think a distribution that automatically includes all the proprietary stuff that we use baked into the distro would be more popular since it makes linux ready to go for most people; but it still gets fewer than 300 clicks per month.

hyperbola

they have a wiki with insane nonsens about why they don't package certain things. Example:

pam
Package has different security-issues and is not oriented on the way of technical emancipation as Hyperbola is trying to adapt lightweight implementations.

https://wiki.hyperbola.info/doku.php?id=en:philosophy:incompatible_packages

gobolinux

it's main feature is that it completely redefines the system's root directory structure. the only reason i even know it exists is because i'm friends with one of the creators

Meego, a combination of Intel's Moblin and Nokia's Maemo. It only ever shipped on one device, the Nokia N9.

Have you ever heard of arch? That's what I use by the way

Suicide linux. Nobody can run it for more than a day

Edit: i just searched "suicide linux" to see if it still exists and one of the top results was ian murdock's wiki page, :(

KISS

it's just a single bash script and a repository containing package definitions to compile them from source.

Basically LFS on drugs.

Rarely hear about DSL anymore, http://www.damnsmalllinux.org/

Take your pick from the Linux family tree

most obscure and to me coolest but unfortunately not very active https://sourcemage.org/

Yellow Dog

I actually ran this on a PPC Mac back in the day

Well I don’t hear much about Gentoo, Damn Small, Puppy or Knoppix anymore. Wonder if they still exist.

I haven’t done much disto hopping since I settled on Ubuntu around ‘08 and then on NixOS last year. I like my systems working when I need them and waiting around for a new install to finish is boring to me.

Bohdi linux. smoll and beautiful. Used to run it on my eeepc 701

Jarro Negro. Made by Mexican students. And as far as I know, it's independent, not based on another distro.

I'm gonna go with Tom's Root Boot. Or maybe the father of all live distros, Knoppix.

Limiting to those I have used daily and treated as Linux (used the terminal for example) probably Maemo. I used to carry my Nokia Internet Tablet 770, and then my N800 everywhere with me.

Maemo is also an ancestor of both Tizen and Sailfish OS

Red star OS

The old PearOS(which looked like a meme-ish knockoff MacOS), UwUntu and Nyarch

Sabayon Linux. I'm not sure if it's still releasing updates, the main website is dead. It was based on Gentoo and later funtoo, but had a package manager of precompiled binaries. You could still use emerge if you wanted to. Definitely a weird and interesting distro

Blend OS is trying to do the declarative nixos thing but with an arch base. That's pretty cool.

ClearOS was Intel's attempt at an immutable os. From what I remember it was really fast.

Edit: actually it clear Linux not clearOS. Edit: also clear Linux is stateless. I don't know, there's a lot about it I don't understand

I imagine there was a time when this wasn't obscure, but I'm guessing people today don't remember Caldera OpenLinux. That was the first Linux distro I installed/used. A guy from church gave his copy.

Caldera eventually became SCO. But I'm pretty sure I was using Caldera OpenLinux before the whole Novell patent suit thing.

Obscure as in "only for a very specific purpose and nothing else"?...

Well, there is the Mircrosoft linux distro for their azure cloud

I guess DD-WRT as distro for router is also kind of obscure. Or the more general openWRT for embedded systems.

There was a bunch of weird rebadged Ubuntu derivatives back in the day.

Ubuntu satanic edition. https://archiveos.org/ubuntu-satanic/

Ubuntu Christian edition. https://archiveos.org/ubuntu-christian/

Hannah Montana Linux https://hannahmontana.sourceforge.net/

I see no one has mentioned Bedrock Linux yet. Not sure though how others would rate its 'obscurity' though. It's definitely a standout among distros.

I haven't tried all that many distros, but I'd say Puppy Linux. Pretty neat that it loads into RAM from USB and has fairly light memory requirements, but it does feel a little on the clunky side as far as configuration and stuff goes.