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

I have a thumb drive with Mint Mate installed on it and it runs fine on a 4gb i5 - 3rd gen.

river or sway

I'd go with XFCE

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

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

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

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.