I did some tests this weekend, I can can definitely keep a Debian or OpenBSD installation happy with 128MB of RAM as long as I stick to a trimmed down version of Xfce or LXDE with Abiword. This wouldn't be my preferred setup otherwise (Slackware, naturally) but you will see why I want an OS that supports multiple architectures in a minute. Unfortunately it looks like Debian has included more junk packages in its default Xfce installation (not as bad as Ubuntu, but still enough to raise the memory requirements) since the last time I tried it. I have been so impressed with Debian and the BSDs in supporting alternative architectures, including VAX(!), older PPC, and even Loongson, which I had only previously read about (I think US export
restrictions on Xeon Phi boards will raise R&D and quality control investments in this chip soon). I have also been impressed with the basic research I have done on FreeBSD jails and will want to do some testing on this for my paid work. But why all the research into alternative architectures and operating systems? My wife is looking for a laptop that would do only the basics for her dissertation writing, and strange possibilities have shown themselves to be very affordable.
(1) Clamshell iBooks (1999-2000) are cheap (usually about $200 on eBay), and the ones for sale look to be in decent shape if you ignore cosmetic details and don't expect a 15 year old battery to work well. They usually have about 128-256 MB of RAM. I like the idea of making vintage computers fully up-to-date and functional, and apparently so does whatever BSD geeks worked on the VAX port!
(2) 2012 model Chromebooks would be substantially more powerful than the iBook, if less visually appealing. It seems all of the current generation of Chromebooks use Intel processors, but the earlier ones with ARM chipsets should offer very good battery life. Looking to dual boot Chrome and a proper GNU/Linux, with an eye to making it functional beyond just Google apps.
(3) Banana Pi + battery pack + touchscreen. Yes, this is a little more out there, but it could be a fun project, too.