Share Internet via Ethernet - how to set IP?

Your mac will act as a DHCP server.

Go to System Preferences->Sharing and set it to share your connection from Wifi to computers connected to Ethernet.

Then in the Linux machine configure it to use DHCP.

That is good advice, Bob.

I agree that a direct connection will in general be preferable.

Only the OP can tell, but one possible case is having an old desktop computer without Wifi, which is located near the mac but far from the router. In that situation, Internet Sharing would be a good solution.

Cool. Now I understand.

Maybe you could extend a usb port to outside the base to use a wifi adaptor. Just brainstorming. You've already figured it out.

https://www.amazon.com/Wireless-Adapter-Extension-Cradle-5-feet/dp/B0091WWNAS

You already got the answer from Luis but I wonder why you just don't connect your second computer to your network using WIFI or Ethernet rather than using your Mini as a router?