What's in a Name?

Question: How does a god gain an Epithet in other cultures? Also, what does it really do? I mean, they get a name for that culture, but how does it affect the god?

Your capitalization of the word Epithet makes me think that you might be playing in a game that has specific rules about those attached to the Fatebond system. I know those games exist, but we aren't one of them, so unfortunately we can't help with those mechanics much. Refer to your Storyteller, if that's the case!

However, if you're just asking about divine epithets in a general sense, like Sowiljr being called Tlazohtlaloni in Mexico or Jioni being called Apostolia in Delphi, then we can talk about that a little bit. Epithets of this kind do not actually "affect" a god, at least not mechanically, but are important tools for Storytelling and character development, as well as being neat things that players get to come up with for themselves.

Gaining an epithet only requires one thing: the Scion has to become known to that culture as a deity or sacred figure in some way. That culture, once they've realized that the Scion is important and has some kind of important function, then naturally gives her a name in their own language, which is usually descriptive of what she does and how they view her. When Eztli's bloody Icelandic rites were established, the people began calling her Ristablodr (Blood Eagle) to link her to them, and when Folkwardr became famous in the Germanic forests for his exploits as a frightening forest-lurking monster, the people began calling him Ungeheuerlichgurtelschildkröte (Terrible Turtle Monster) as his proper name there. This is something that usually happens organically as a result of what the PCs have been doing in various areas, what kinds of effects they've had on the local people and how those local people can fit them into their pre-existing beliefs. It's also possible to get epithets that don't apply to one of Scion's religions, as one PC did when she accidentally became declared an angel in the Greek Orthodox Church, or when the three male PCs from the Better Next Time game became the Three Satans of a new church dedicated to opposing them.

These epithets are a lot of fun for players, who we encourage to explore what they might be when they're getting close to hitting godhood; taking a minute to look back over their Demigod careers and see where they've already made some new history and religion is a good time, and they get to have some input into how they hope they'll be seen by cultures outside of their own. They can also give the Storyteller some good idea sparks when it comes to plots that might affect the PC in those cultures, which is especially helpful when you have PCs from different pantheons and need to have a way to make a story set in one culture still relevant and interesting to PCs who aren't native to the area.

However, mechanically, epithets don't do anything. They're names and representations of your young god's religious importance in various places, but they don't actually affect her stats or abilities any. Remember that mortal worship does not directly affect any gods in Scion except for the Teotl, so having more people believe in you, regardless of what culture they're from, doesn't actually do anything to your character.

But, being believed in by various cultures is often a by-product of having a worshiping cult there, which can affect a young god by way of Fatebonds, so in that way how they think of you and what their image of you might be is enormously important! It's because of Sowiljr's Aztec cult that worships him as Tlazohtlaloni, for example, that he has Water Fatebound to himself as their god of ponds, and that Fatebond shapes him over the course of time.

Really, it's just one of the cool ways you get to explore being a god and what that means to various people you've interacted with and affected. Sometimes it's good, sometimes it's bad, sometimes they're really weird and buy you a bunch of Control (Landscaping Truck), but it's something that illustrates what you're already doing in the game, not the other way around.