In order to use homebrew and ethernet on the Wii Mini at the same time, a USB hub is required, as the Wii Mini only has one USB port, and there is no other way to load homebrew on it.
We recommend purchasing a powered USB hub with a built in ethernet port, such as this one. If you do not get a powered USB hub, the console may not be able to provide enough power for all your USB devices.
Once you have a USB hub, you may proceed to enabling Ethernet on your Wii Mini