That's a weird problem. I've never seen anything like it on a computer that isn't scratching for resources. Still, a few things come to mind.
First, if you exit X and go to the terminal, does the problem still occur? If so, then ignore the next two suggestions. Otherwise, for where I talk about editing files, you'll need a HDD install. If running from CD or a "frugal" install, just do a "killall" command e.g. "killall torsmo" and see if it solves the problem (not quite as certain as disabling it at startup though).
--edit: Or just exit X, edit the files in the RAM drive using nano and then do "startx" to bring up X again. If you've run System Rescue CD, I'll assume you're happy with the Linux terminal.
That resource monitor (torsmo) at the top right of the desktop updates I think at the rate your problem it happening and it includes a network monitor (which has never worked on any system I've used with the net). The program can be disabled by commenting it out of the file "/home/dsl/.xinitrc". If that fixes things and you still want it running, you could try editing its config file "/home/dsl/.torsmorc" to disable just the network monitoring function.
You could also try changing the window manager to fluxbox and right clicking on a desktop icon and choosing to exit DFM from the menu (or to do on startup, edit "/home/dsl/.desktop" to read:
If all this fails (not unlikely), it might be time for a bit of research. If your laptop is a rather unusual modern design, the chipset it uses might not have been fully supported in the DSL kernel (version 2.4.31, the other distros you tried use much later versions). You might have to hit Google hard to find out. If so, then there may be a driver package that you could compile for DSL and load as module/s on startup (however it may also exist and not support kernel 2.4).