The settings in the Network Preferences panel, in the System Preference selection area, can be changed and changed back, some are saved in the system and you can switch back and forth. WEP is a weak 'wireless equivalent' security protocol so it seems odd that it'd be in use nowadays by anyone, but it still is, by some.
There is a way to change or save new Locations in the Network Preferences, and likely if the computer has been used via IEEE ethernet (cable or wifi) the owner may have used the 'MAC address' or machine ID, in conjunction with those settings to make a new secure Location for use at home, otherwise he'd have to do a bunch of stuff over and over again. Senseless, it remembers them under different Location names. So when he travels with the computer, a different set of Location is chosen in that Network panel. And if you approach a library network with a passord protection, they can tell you the PW and it can be entered into a Public location. Some do not require one. So they are less secure because they can be open to people nearby using their computers to look for unsecured visitor devices. And each location may invite use of a password, even if the hardware is sitting right in front of you, but set up by someone else, to insure their secure motives.
The direct by wire without a MAC address should be a matter of settings in the Network Preferences, such as TCP/IP and so on, look around but don't change anything. Sometimes the alternate settings are already there, but not chosen after traveling with a portable.
I've thought of several answers, researched a few others, and decided to not go further with this. After all, the classic 'Forbidden Planet' is on TCM again, and it is rather distracting, yet more honest than talk radio.
Hopefully someone will point out all the shortcuts a frequent user of the Network Prefs & setup of MAC address connections. When you used the WEP, you used a password to access the local wireless, so if you make and save a new Location in the G4 and point to the router, set on Automatic, it should just work. And that may stop it from looking for the Ethernet. {But any breach of security in his computer would be on you.} See the Airport icon in the main menu bar and click on that, to see about 'join another network' if you have a password for WEP, if you can find the correct one by name, or make new one in Network Prefs.
PS: find the computer build information and original OS X
by use of an 'identify by serial number' site such as this:
http://www.powerbookmedic.com/identify-mac-serial.php
Good luck & happy computing! 🙂