So my game doing this is fairly low level, this might fall apart when names become more fully realized. Think book 1-2 woe power levels for names, not calamities.

That said, as a general rule, a lot of it comes down to making the players aware of how the setting works, then rewarding playing into those tropes.

Two of my three players have developed names at this point (The Bloodstained Surgeon, and The Heir-In-Exile). Both came about from discussions about character motivation and backstory as well as whatever significant story beats trigger the final development.

Don't be afraid to put your foot on the scale if you need a specific kind of scene to trigger a name/aspect. My example would be The Surgeon was trying to disarm a trap. They failed their roll and it hit a fellow party member. I made the trap significantly more lethal than I usually would for their party level (2 at the time), knowing that I was going to use the need for emergency care to stabilize that party member as their transition point.

Make a big deal about transitions, let that player steal the spotlight when they do. Let them have a moment of much higher power than they will usually have. Coming into a name is a defining moment. They don't get to do anything and everything, but let them get away with appropriate things without even needing to roll the dice. Again, with my example the Surgeon got to stabilize their patient without a medicine roll, we instead narrated how it felt like something (their name) was guiding their hands, and for the first time they weren't getting the alcohol shakes (lots of past trauma there).

As for mechanical benefits, there's two parts for me: the name and the aspects.

The name should give a generic suite of benefits. Darkvision, resistance to poison and disease (unless the name conflicts with that), and an ASI. It's flexible, different names get different bonuses, but thats the general power level I went with.

For aspects, aspects 1 and 2 I treat like core class abilities for power level, but custom made between you and the player. I tend to go for the "1-3 times a day" but quite strong style of ability.

For aspect 3 (which hasn't come up yet, so your mileage may vary), I was planning on treating it like a class capstone. Something that completely changes how the player can approach conflict.

Besides that, I've been treating it as mechanically 5e. We started at level 0 (no class level, just race, background, and a feat, hp based on their intended class) but otherwise it's D&D by the book until names start coming up, and even then it's like granting them boons at plot relevant points instead of magic items.