If you can't hit the server select screen on those days, are you able to access the website
http://eqemulator.net?
There is not something physically wrong with the connection or modem. If this is not down your alley.. I would start off by pinging eqemulator.net when you have the problem.. That is the login server. If that resolves. Try pinging the p1999 server IP in the eqhosts file. If that works, you are actually reaching the server via ping. At that point, it's a port/router issue.
The grand test would be trying a new router or hooking your computer directly up to the internet. Sounds like hooking it up directly to the net is not an option, so I would be looking at the router. EQ uses a LOT of ports. But if you can't get to the login server, that's a pretty simple request.
Him playing WoW should have absolutely zero effect on your EQ.
Meant to say.. If this is NOT down your alley.. Find the geek. Tell him the server works fine, always. Give him a case of beer. Do your troubleshooting on a day where it's not working at all.
Me? I would troubleshoot it. Turn off software firewall programs, remove them completely from your computer if necessary.. Uninstall antivirus if necessary. Burn the computer down to barebones until EQ starts working... then work your way back up. Nothing on the PC working? Buy them a new router that's better than the current one.. Have the geek implement it.