Can a Mac Mini extend my Airport Extreme wireless network?

never heard of a net card for osx or windows or linux pc's which had the feature

In theory, yes it would work. But is it practical? Not really... internet sharing with a Mini gives network devices a different IP address than your Airport which may cause interruptions and issues with the handoff.

The mini doesn't have that powerful of a wifi card, so overall I don't recommend it.

Get another Airport, Asus, or Netgear wifi router, put it in bridged or access point mode with the same SSID and password you currently use and boom! your network is now extended!

Make sure to set your channels properly 1,6,11 on 2.4 and with 5ghz, with so many channels, try the high or low-end for channels (see photo below as a reference) Green is channel 153 and Orange is channel 157, so every other channel should be ok on the 5Ghz band for performance.

Use Netspot to determine which channels in the 2.4/5Ghz range are best for your area.

This is the best practice in the industry, ethernet all AP's, no extensions or repeaters, map out which channels work best with minimal overlap.

Also, Internet Sharing would require the Mac be connected tot he Internet via Ethernet in order to share the internet connection via wifi