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While Renown is a measure of your Dynasty's overall notoriety in Crusader Kings 3, Prestige indicates your character's personal fame. Characters with high Prestige (also called Fame) are well-respected among the nobility and will have an easier time getting their way. Prestige also increases your Fame Rank, allowing you to wage more ambitious wars with greater rewards.
High Prestige is always useful, and unless you act dishonorably it's generally yours to keep once earned. Try using these methods to increase your Prestige quickly, especially if your character is a young adult and needs to prove themselves to their peers.
Win Battles And Wars
Rulers whose armies are victorious on the field of battle gain a Prestige boost from each win. If you have the money and the troops for frequent wars, this is a sure-fire way to earn Prestige. You can get even more Prestige by personally leading victorious armies, but this should only be attempted if your character's Martial and Prowess stats are solid - dead characters can't earn Prestige, after all.
Holy Wars tend to generate Piety rather than Prestige. If you're looking to increase your Fame Rank, stick to secular wars.
Wars can also earn you Prestige in the long term. You gain Prestige every month based on the amount of land you control, so conquering more land increases this gain. Furthermore, if your character becomes wounded in battle, they may gain the Scarred trait once healed, giving them a small amount of monthly Prestige for the rest of their life.
Have Prestige-Boosting Traits
Scarred is one of a handful of character traits that grant extra Prestige. Arrogant and Impatient characters earn more Prestige every month, but these traits carry a number of negative effects as well. In particular, a character with both these traits isn't likely to have a very impressive Learning stat.
Characters with a Martial Lifestyle looking to earn extra Prestige should complete the Gallant Skill Tree. The available perks are handy for winning battlefield glory, and the final skill gives your character the Gallant trait which boosts your monthly Prestige income by a whopping twenty percent.
Marry From Renowned Families
When the time comes for your character to get married, choosing a bride or groom from a highly-respected Dynasty can give you a Prestige bonus. This is especially true if your intended is the relative of a Royal or Imperial monarch. If your Religion permits your gender to have multiple spouses, you can do this several times over for quick Prestige gain!
Make Prestigious Choices
Your character will see dozens of event decisions over the course of their lifetime. Often, taking a choice that asserts your authority or shows your noble spirit will come with a healthy Prestige boost. You can always see the results of an event choice by hovering over it, so check all the available options before making your decision.
Hold Frequent Festivities
Parties such as Feasts and Hunts can be expensive to hold, but each time you do you'll gain Prestige. These events can also give you new opportunities to make friends and influence your fellow nobles, making them a win-win.
Feasts are generally more beneficial to Diplomatic characters, gaining Prestige by spending lavishly on your guests and schmoozing when they arrive. Hunts are better for Martial characters, allowing you to show off in front of other nobles by bringing down dangerous beasts.
By default, notifications for Feasts and Hunts are disabled, but if you intend to make them frequent occurrences you can check the box in the Decisions tab to be alerted as soon as they are available. Wealthy rulers who can afford to do both every five yearsstand to gain a lot of extra Prestige over the course of their lifetime.
Set Your Heir Up For Success
Eventually your character will die, and you'll take over playing as your heir. One of the best ways to ensure a smooth transition is to make sure your heir is gaining Prestige while you're still alive. That way they'll have better relations with their vassals when they inherit.
As soon as your heir comes of age, try to find them a prestigious spouse to get the ball rolling. Once they're married, grant them some land (ideally a whole Duchy if you can swing it) and maybe give them a Council position. It's important to wait until after your heir is married to give them land, as once they are landed they can choose their own marriage (which you don't want).
If your heir has a high Prowess, make them a Knight and send them into battle. This carries risk, of course, but will quickly earn them extra Prestige. If you decide to send your heir to war, try to similarly prepare the second-in-line, just in case.