There is no activation process with Lightroom. I license entitles you to two installations for your own personal use. Just install on the new Mac mini.
There is no deactivate process for Lightroom.
The Lightroom License key is
a. Single owner / user, and is cross platform Mac & or Win.
b. The agreement allows two active installations with one in use at any point in time.
c. You can install / uninstall as often that is required provided you respect the above.
d. You just install and use the License key to activate.