Debian on old or slow computer
Consider using OldStable instead of Stable. Add Backports to your repos to get more software. Debian 6.0 (squeeze) will get a special Long-Term-Support (LTS) for i386 and amd64 until February 2016. Some packages won't be supported. Links: LTS/Development and CategoryLts.
Replace websites with special software
claws-mail (former Sylpheed) is a small mail programm with the look'n'feel of thunderbird/icedove. It features pop3, imap, smtp, spam-filter, encryption and plugins. clawsker is an extra utility for configuration.
dradio is a terminal based frontend for mplayer. It plays the preconfigured danish radio and TV channels. You can configure your own channels in ~/.config/dradio/menu.xml . The config file format is described in dradio(5).
pinta is a small, but powerful paint.
streamripper rips audio streams into files.
dwm epdfview dvtm (on desktop) screen (on server) irssi lftp gscan2pdf rtorrent scrot qalc pydf w3m (with w3m-img ) mosh (ssh for slow or unstable connections) iceweasel with plugins jave (not in repos)
If you can watch small videos only, recode videos with: nice -n19 ionice -c3 mencoder -oac lavc -ovc lavc -lavcopts acodec=mp2:vcodec=mpeg2video:abitrate=128:autoaspect -of mpeg -vf scale=360:-2 -o small.mpeg big-source.avi
Make use of ionice (package util-linux). The CFQ has to be the active io scheduler in kernel. Example: nice -n19 ionice -c3 aptitude update