Lately I've seen a few people visit my website by searching with Slackware & Eee PC-related terms, presumably to see how well Slackware 13.37 runs on netbooks.
Good news: It runs very well. I've spent over two months now, using an ASUS Eee PC 1000HA with Slackware 13.37 as my primary computer while my ThinkPad's technical difficulties are sorted out. It's been pretty comfortable.
The Slackware installer detected all the system devices correctly, including Wi-Fi, so everything worked out of the box. KDE 4 was running pretty well, but I opted for Ratpoison, plus some other tools as my desktop environment. System performance is pretty good. I haven't had issues with RAM, although I upgraded from 1GB to 2GB when I bought the machine. The biggest bottleneck is the CPU, especially when compiling something, as I so often have to do when using Slackbuilds. My netbook is two or three years old now, so I imagine newer models will perform much better.
I'd say that Slackware is a great choice for your netbook, as long as you're into advanced Linux distributions. The KDE desktop environment may be a little slow for netbooks, but Slackware also comes with Xfce, Blackbox, Fluxbox and Fvwm out of the box. Pick your poison. :)