"ifconfig eth0 remove" or "ifconfig eth0 destroy" Or you can open the Network Preferences pane on the mini and select "release DHCP" for the ethernet connection. All of these should release the IP. But that's not your problem. Your problem is that the DSL modem seems to be remembering the hardware address of the mini's ethernet card (it's called a Media Access Controll address, or MAC address), and not assigning the PC an address because of that.
It sounds like you could use a cheap cable/DSL router. Not only would you no longer have to worry about swapping cables around, you could surf with both computers at once. If you shop you can find these things for around $30-40. The neat part is that you can tell these routers to report any hardware address you want. So pop down to CompUSA, buy one of their house-brand routers (or any name brand router for that matter), and set it up. The configuration for each brand is different, but they all support "MAC Address Cloning." Plug in the address for your Mac, and you should be golden.
Another way to get around this-though it's a hassle-is to turn off the modem and let it sit for a few minutes, then plug it into the PC. When it starts up again, it should connect to the PC. The trouble is that you'll have to turn it off again when you want to connect to the mini. A router is the way to go.