County of Fortune how to choose a mayor: open the settlement window, click the empty red portrait beside the mayor's name, then click the portrait of the mayor you want. Look at the bonus before you commit, and remember that a bonus you cannot use yet is not a bad choice.
County of Fortune how to choose a mayor: the steps
Open the settlement window, click the empty red portrait next to the mayor's name, and click the portrait of the candidate in the list. The same window is used later to swap mayors. Every candidate belongs to one of the noble houses, and noble houses covers what that house brings with it.
Reading the list before you hire
A mayor's bonus is tied to a particular industry, so the useful question is whether the county has that industry yet. The current selection window merges new candidates and the mayors you already own into one list, so you can compare hiring somebody new against moving an existing mayor. A row of industry icons filters the list, and each candidate shows which of your settlements would gain and which would lose if you hired them.
Swapping a mayor between settlements carries no penalty at all, so moving a specialist mayor to the settlement that actually produces what they boost is a free decision. If the industry itself is the problem rather than the mayor, changing production covers replacing the building.
Why there is no perfect mayor
Mayors are not a pure random draw — the game generates the bonus that is least common at the moment. It is still not built around giving you a perfect mayor for every building, and you are not supposed to have every industry matched with an ideal mayor. A mayor you cannot use right now is still useful, because you can always build the matching building later and put them to work.
Since the rebalance, negative mayor bonuses have been removed altogether, so a candidate is always a net gain somewhere; what remains is deciding where that gain is worth taking. Every mayor belongs to one of the noble houses, and the weight of that house matters as much as the bonus itself — noble houses covers power, relations and elections.