Legitimacy comes from Ambitions, Cognomen and random events. Each point of Legitimacy grants 1 Family Opinion and 0.1 Orders per turn.
Ambitions are a Leader's desires for what they want to achieve. Completing an Ambition will give +10 Legitimacy while failing or abandoning an Ambition an Ambition will give -5 Legitimacy. When a Nation Leader dies its Ambitions become Legacies, meaning that they will fail if not completed within 20 turns. Legacies provide only +5 Legitimacy when completed.
Legitimacy gained from events and Ambitions does not decay when a new Leader takes the throne. It is better to complete Ambitions before they become Legacies.
If a human player Nation completes 10 Ambitions (including Legacies) an Ambition Victory will be achieved if this Victory type is enabled.
Cognomen
Gaining a new Cognomen
The Cognomen is the title that appears after a character's name and is granted after a character does enough specific actions. Cognomens affect Legitimacy and once a character acquires a Cognomen it can only acquire a different one if it grants more Legitimacy than the current one. If a Nation Leader dies the Heir will start as "The New", which grants no Legitimacy.
Legitimacy from the Cognomens of previous Leaders decays each time a new Leader takes control (it starts at 50% and decays to 33%, 25%, 20%, 16.6%, 13.3%, 11.6% and 10%).
The renown threshold for unlocking Cognomens increases by 5% for each subsequent Leader, making them more expensive as the game progresses.