If you want rock-solid stability and security, try FreeBSD (I think their most stable supported version atm is 5.4), but bear in mind you're going to have to work a LOT harder to get it working than you would for, say, Gentoo (I run both - Gentoo for desktop/laptop from stage-1 installs, and FreeBSD 4.10 for my server... which has been running without a reboot since February '04 with frequent updates).
For linux distros, go with Debian or Slackware if you want something relatively easy to configure... or if you are linux-savvy you could try and tackle Gentoo (attempting to do so if you're not is an exercise in futility). Any of those either have packages for what you're looking for or can compile all those packages no problem.
If you want a graphical desktop over your network (you don't need a monitor), install tight-vnc... though I recommend you don't allow it to accept connections from the internet. What you see if you had a monitor and a graphical desktop is what you get on the other side, with whatever OS you use (even Windoze).
As for Samba, I haven't seen what Flare was describing... I've used it on Mandrake (older versions) and more recently on FreeBSD and Gentoo. I've also used it to automatically install the correct printer drivers on every windoze machine connected to my network (no configuration on the clients - it just shows up).
I've never used eggdrop... I've always been an Apache2 guy, and 64M was more than enough memory to handle that.
<OT>Heck, my original webserver was on an old P-200 with 64M EDO RAM running Mandrake 9 and using software RAID-5 across 7 slow and ancient SCSI hard drives (1G each to make 6G space). It was a very poor choice of distros and it took several days to install (menu problems and just general inefficient code on both Mandrake and Fedora's part), but once installed everything worked fine (and sped up 10000% from the install, too). That box was an Apache2 webserver (with .cgi support - in fact its main purpose), ssh server, vnc server, and last but not least a small print/file server (all critical files went there for the RAID protection).</OT>