What distros have you tried?

While Linux is certianly capable of being very lightweight, it really depends on what all you have running. One mistake that the *buntus make is that they automatically install and run many services like SSH, Apache, and others in the background when 90% of the users will hardly know that they are there, much less use them.
For example, right now I am running Debian Jessie on an older machine (I don't know details, I just rescued it from the trash. Intel's website tells me that the processor is from 2006). I use LXDE, a lighter desktop environment, and even with Firefox running (as well as several services, such as SSH and Apache), my RAM usage is hovering a hair under 1GB and my processor usage is in the single digits.

There are plenty of lightweight distros capable of running in as little as 64 MB of RAM even today. Mostly it comes down to what desktop and applications you use, really. And what runs in the background matters too. In a Linux universe nothing is compulsory, and you can have a running system with nothing but the kernel and bash. Take any distro, do a minimal install, and then add on stuff you want. It's the best way to go about making a light running system.