Select the empty red portrait in the settlement window
To appoint a mayor in County of Fortune, open the settlement window and click its empty red portrait. Then select the portrait of the mayor you want to appoint. This assigns the settlement's leader and lets you choose a production bonus that suits its role.
Those are the appointment steps given in the official County of Fortune demo FAQ.
Compare the bonus with the starting building
Before selecting a mayor, look at the bonus and the building your settlement starts with. The FAQ recommends choosing differently depending on that production.
For example, a settlement built to support a crop chain has a different production role from one containing a guild or equipment workshop. Use the displayed bonus to judge whether it applies to the work you want that settlement to do, rather than assuming any impressive-looking percentage benefits every building.
You do not need a perfect match in every settlement. The developer explicitly says it is fine to have a mayor whose bonus does not apply to the buildings you currently have, because you can swap them later.
Compare available leaders in the updated window
The October demo's mayor rework brings new and existing mayors into the same selection window. It is designed to make hiring and moving an existing mayor easier to compare.
Negative mayor bonuses have been removed, along with the penalty for swapping mayors between settlements. If an older demo guide tells you to preserve a poor production match solely to avoid that old swap penalty, check it against the current rework.
As a settlement changes, revisit the comparison. Adding a new building or replacing the original production can make a different mayor a more useful fit. The appointment system is not a requirement to keep the first selection forever.
FAQ
Where do I appoint a mayor?
Open the settlement window, click the empty red portrait and then select your chosen mayor's portrait.
Must the mayor's bonus match the starting building?
It is useful to compare them, but the official FAQ says an unmatched mayor is acceptable and can be changed later.
Can I compare a new mayor with one already serving elsewhere?
Yes. The October rework puts new and existing mayors together in one selection window so you can compare a hire with a swap.