Hiya!

I use a smallish router (courtesy of my ISP) in my
home network. A few times this past week, it has
refused to give one of the computers an IP
address (It's set to run a DHCP service).
This computer (running Mandriva) says
"martian destination 0.0.0.0 from 192.168.1.254" (Note: Router IP)
at bootup.

A reboot of the router solves this problem, but I'm looking for a
more permanent solution. Since routers aren't capable of selective
hate, I'm guessing it's something to do with the settings on the computer
itself. Any and all ideas are appreciated, as usual.

0.0.0.0 seems like it's sending a broadcast packet. I'd run tcpdump or ethereal on the workstation to see exactly what the packet is and get a better idea of why it's sending it. What kind of router are you using?