I'll add that it's helpful to play the short and the long arcs of characters.
What I mean is that, in addition to the role-playing aspects of a character in the context of a single event, conflict, or short-term goal, there are also the the seasons of a characters life and how that may influence which lifestyles to prioritize, shift goals, or otherwise.
For example, I played a character obsessed with chasing the action and foreign adventures abroad in their youth (starting with joining William the Conqueror in exchange for a profitable marriage). As time went on, and injuries began to stack up, and a near death experience turned their attention to the church. This lead to a lifestyle change as well as the granting of lands to theocratic characters to the point where they became the greatest power in the realm (a problem for the next generation); unable to crusade directly in later years, their driving goal was to then support the founding of a holy order which came to equal the prior theocratic holdings. Of their two sons, the older was raised by the leading clergymen of the realm, while the younger was trained by the grand master and returning crusaders; they became, in essence, the two sides of the starting character's opposing nature, to be concluded by the fraternal rivalry fueled by the two great political factions of the realm...
If you're not playing an ironman game, try swapping characters within your family, realm, or region to pick up on the more interesting story threads.
For example, if your character loses a war to a character, you could arrange to have the character you lost to (or someone in their court/realm) act as your heir's guardian and then swap to play that new guardian character. This way you can still follow your primary dynasty's story from a different perspective. In the same vein as CrazyMe suggested, this would work great for travelling with characters (crusade, marriage, etc.) to other parts of the world as well.
If you zoom out and see something particularly interesting going on in the world, you can always tie your story into that or not; maybe take a break from your primary dynasty characters a while to play as the Mongol invasion for a time as they creep closer to the Crusader kingdom one of your relatives reigns over?
Lastly, leaving the game to read about the time period or do something completely unrelated can help keep the game feeling fresh when returning to it.
Of course, it's ultimately up to you and your imagination to make the most of the sandbox.