Testing an Ethernet peripheral
I basically just plug it in and have it do a request, make sure the data gets through OK.

Test heavy data transfers (or at least as heavy as the design will be rated to handle), try as many different hubs/switches as you can (10, 10/100, Gbit, etc). Try different duplexes, etc.

In my experience, if the design is sound and one unit verifies OK, the hardware is not the likely point of failure - issues are more likely to crop up in the TCP/IP stack or application code.