Nicest distro I found so far for weakling hardware is LXLE http://lxle.net/. That's a better, fancier version of Lubuntu. Alternatively, if you wanna break away from the world of *buntu how about a Arch-based LXQt distro? https://antergos.com/ That stuff is like superduper speedy.
Mint and *buntu are basically the same, only difference is that Mint comes with ralleystripes, fairydust, goldglimmer and codecs and non-free drivers. Oh, and they make the most lovely implementatios of the DEs, as one can see when you compare Mint/MATE to Ubuntu/MATE. Mint's version is so much betterer and nicerer.