I was having a really tough time getting things going until I found an article on adapting Ubuntu to low memory situations at Ubuntu’s community documentation site. The basic idea outlined there is to install a text-based system and then manually add lightweight GUI stuff and other applications. One cool thing about this is that we don’t need to limit ourselves to an Ubuntu based install: since Ubuntu is itself based on Debian, it should be possible to do with Debian what the article suggests doing with Ubuntu. The article also mentions the possibility of installing Xubuntu and then replacing the heavier items with lighter alternatives.
So, overall this gives us three different promising paths to follow:
1. A text-based Ubuntu install plus a manually installed GUI environment.
2. A text-based Debian install plus a manually installed GUI environment.
3. A full Xubuntu install with unnecessary stuff removed and heavier items replaced with lighter alternatives.
Approaches similar to the above should also be possible using any other major distribution. However, at this point I want to limit myself to Ubuntu and Debian. These are two very well-supported distributions, and I don't see what will be gained by considering other distributions. Of the major Linux distributions, Ubuntu is the most consumer-oriented. Debian is an attractive alternative to Ubuntu because Ubuntu has a reputation for doing some things in a slightly non-standard way; Debian is an accepted, established, and carefully maintained standard that may overcome some problems presented by using Ubuntu.
I have been working mostly with parallel installs based on 1. and 2. I will postpone working along path 3. until after I have a better understanding of what’s going on. An additive approach to functionality lets you learn what each package, etc. does and gives you a more fully integrated idea of what’s happening. A subtractive approach only tells you what’s not needed—and it doesn’t guarantee that you have a minimal set of what is required. However, I will probably play around coarsely with an Xubuntu install before I have a really good idea what's what to see what might be possible along that path.