Why not Arch? I used it a year ago on an old hand-me-down laptop I got from a friend and it wasn't very resource intensive. You could also try other arch based distros like Manjaro but I have no experience on that end.
Manjaro is pretty much a pre-set up version of Arch, so if the OP isn't down for a vanilla arch install, that would be the way to go.
But anyways, I had a similar quest a long time ago back when I had an Acer ZG5 with 1GB of ram and a single core 1.6GHz processor, haha. I ended up settling for Crunchbang linux as it was debian based (a.k.a. not bleeding edge, but super stable), but that is now dead. In it's place there is Crunchbang ++ and Bunsenlabs Linux, both of which are pretty good. But the best part of CB was how low resources it was. It didn't run a desktop environment and instead ran something called OpenBox which was just a window manager. This meant that with my 1GB of ram, the system was barley using 200MB on any given day. Super nice.
Another alternative on the Linux side of things would be Puppy linux, which loads the entire system into RAM, making the system super snappy. It kinda looks dated to me, but then again, I haven't used it since the days of Ubuntu 14.04, so what do I know...
If you are uncomfortable with linux, you could give Chrome OS a shot on your system through CloudReady. It'll completely wipe your drive, and it is very unfriendly if you want to dual boot multiple OSes, but Chrome OS itself is very light and very capable. Keep in mind though, you can run most windows applications on Linux through a compatibility layer called Wine, but on Chrome OS you are pretty much stuck with chrome-os applications only (I.E. botched android apps, web apps, stuff from the chrome web store).
Elementary OS is still a lighter linux OS, but it is still based off of gnome, which isn't exactly lightweight. Remix OS is also a fair option, but keep in mind that it is still heavily in development and more of a POC (in my opinion) at this point!
You could also check out Windows 10 for your machine, which is supposedly a lot lighter than Windows 7, but your mileage may vary. I'd also recommend researching your machine and seeing if you can upgrade the ram. Sometimes a simple ram boost of 512MB-1GB can make your computing experience oh-so-much smoother!