I am narrowing the list of possible culprits in my continuing network setup problem.  It appears that the router is working correctly, and that the wireless and PCI network cards are also working correctly.  Here is what I can successfully accomplish:

I can browse the internet from my upstairs photo lab computer, via the wireless card upstairs and the router/access point downstairs in my office.  So, the two wireless portions of the network appear to be working together fine. 

I can browse the internet from my downstairs office computer, via the PCI card hard-wired to the router/access point.  So the hard-wired portion of the network seems to be fine.

From the office computer I can successfully ping the photo lab computer by its IP address, but when I try to ping by its computer name (PHOTOLAB), it fails.

From the photo lab computer I can successfully ping the office computer by its IP address, but when I try to ping by its computer name (OFFICE), it fails.

Office computer = Windows XP SP2 Media Center Edition (XP Pro without the ability to join domains).

Photo lab computer = Windows ME

I have run the Home Networking Wizard on both computers,  and have verified that both share the same workgroup name (MSHOME).

File and Printer Sharing for Microsoft Networks is installed on both systems, and activated on both systems.

Both network adaptors are running TCP/IP protocols.

Client for Microsoft Windows Network is installed on both computers, and is the primary windows logon for both computers.