Here are my current recommended practices for raising a happy skinny penguin. Please note that I am not a Linux expert. That's not such a bad thing because it means I have to deal with all the problems and frustrations that you would from a perspective that is something close to your own. With any luck, you will be able to benefit from my experience in getting skinny penguins going.
Before we start, there's something you need to be aware of. Full-blown desktop Linux (e.g., Canonical's Ubuntu) may have come of age--meaning that it is now possible to run a general-purpose Linux system on recent hardware that is easy enough for anyone used to Windows or Mac OS to configure and use. However, if your goal is to try to install a desktop Linux system on a circa Windows 98 machine, it's not going to be as easy as using Windows or Mac OS. Depending on the route you take, the initial installation may be easy or tedious. But more importantly, after the installation you may need to learn how to configure stuff that Windows and Mac OS do for you without your having to think. This is especially true if you want/need advanced features such as wireless networking, local-network file sharing, automatic mounting of removable media, and the like. Even something as basic as creating a new user account in lean Linux can be a pain. This learning curve can be very discouraging, so unless you are prepared to embrace a lot of stuff that may sometimes have you pulling your hair out, you might want to cut your losses at this point and turn back.
If you are still with me, then read on.
I have broken things down into different categories depending on how fat your penguin is allowed to be. Pick the category that comes closest to describing your situation.
You have 192MB or more of RAM
- Fedore Xfce-spin (untested)
Debian also offers a version with an Xfce-based desktop environment. The Debian-Xfce install is a maybe a bit less demanding of system resources than Xubuntu--but it's also less feature-filled. If Xubuntu is a bit too sluggish for you or if you just have to have a Debian system, try Debian-Xfce. Start by reading the tips described elsewhere on this site if you want to go this route.
You might also consider the Fedora Live Xfce Spin, which I have not explored any further than loading up the live CD and using it for about one minute.
You have between 64 and 192MB of RAM
- Lighten up a Debian-Xfce based system
- Absolute Linux
If SkinnyDebbie doesn't float your boat and you don't mind getting down and dirty with the installation, follow these instructions (including the Even Lighter Debian step) to give yourself a light and fast Debian system that also introduces you to the joys of Linux administration. It's actually kind of fun in a perverse sort of way. Regular Debian-Xfce and Xubuntu will run in less than 192MB of RAM, but I won't say the experience will be a good one. In fact, this whole skinny penguin adventure got started because I was frustrated with Xubuntu 6.10's performance on a 192MB laptop I own and really annoyed by its performance on a 128 MB desktop.
Another option you may consider is Absolute Linux. Absolute Linux is a Slackware-based distribution that uses IceWM as it's window manager and includes many custom utilities to make configuring things easy. It comes with a wide range of applications, and more are available on an additional CD. It's more complete than most other "light and lean" systems I've looked into, and it works. (Or at least it used to work. I've had the version 12.1.0 beta8 of Absolute fail to properly launch a X server session on at least one machine. I hope they fix this.)
The biggest downside to Absolute Linux is that since it is Slackware-based, installing programs that are not part of the original distribution will be harder than would be the case for SkinnyDebbie, Debian-Xfce or Xubuntu. This is because the standard Slackware package manager (the thing that handles the installing of new software) does not do dependency checking--as do most other Linux package management systems. Possible solutions to this include slapt-get and/or swaret--tools for Slackware that manage dependencies and that you might be able to install under Absolute. Also, Slackware repositories aren't as rich as Debian's and Ubuntu's. However, for most users, (e.g., your grandmother) the applications that come with Absolute Linux will be all that's needed, so this won't be a huge deal.
You have less than 64 MB of RAM
- Damn Small Linux
- Puppy Linux
Others have recommended Puppy Linux as well, but I have not tried it so I am unable to comment.