Linux

That's a reasonable machine. You probably could use anything but if you want lighter weight you could use Xfce4. If it is a laptop you could use stock gnome with some swap as a backup to prevent OOM

river or sway

For something with that little memory, I would use a minimal window manager; you'll want every megabyte of memory if you want to have any chance at running something like a javascript-capable browser without constantly hammering swap. fvwm, cwm, jwm, and ratpoison are all small window managers I enjoy; but do your own research into what window manager is the best for you.

In my experiments with a similar setup and integrated graphics, full-wayland Kubuntu feels much more responsive than Xorg-Lubuntu, for what it's worth

I recently bought netbook on AMD c50 for 20$ and firstly, i bought some ram and ssd, luckily ddr3 is very cheap, one or two 8gb sodimm modules and 256gb ssd, or in my case 360gb because price was the same when i ordered them, 360gb was even slightly cheaper, so what i was trying to say, this small cheap upgrade will make a world of difference, and when they'll arrive I'm planning to install "tumbleweed kde" , whole cost of upgrade is 8$ for one module of 8gb ddr3 sodimm, and 17$ for 360gb ssd, 256gb price was the same as i said before

Moksha Heck, just install Bodhi Linux 7, your choice between Ubuntu based or Debian based.

Ironically, ChromeOS Flex would run smooth on those specs, since it does so on my dogshit Samsung Chromebook 3 with shittier specs.

If the PC has an SSD, install anything you want, the PC will handle it fine.