What effect does the patron deity and name day have?

[Feronar]

I'm returning after a while away, and I've always wondered: When you create a new character, you are prompted to select your character's name day and patron deity. What effect does that have in game?

[MistakeNot]

I'm returning after a while away, and I've always wondered: When you create a new character, you are prompted to select your character's name day and patron deity. What effect does that have in game?

Currently: No effect whatsoever.
It used to be that your choice of patron deity had a tiny effect on your Elemental Resistances, but Elemental Resistance as separate stats was removed with Patch 4.2

[Kurando]

Also I think it might make an appearance during the Ceremony of Bonding. It's been a while since I actually got married, but I'm sure the symbol of the two dieties of the bonding partners show up.

[Hasrat]

It... shows up under one of the tabs when you open your character window. That's kinda about it.

[DarkMasamune]

Would've been fun if we received a present in the mog mail when your 'birthday' arrives

But really I don't think it has any use for now, as mentioned by MistakeNot, it used to influence elemental resistances, but it has been removed since.

[Enkidoh]

Also I think it might make an appearance during the Ceremony of Bonding. It's been a while since I actually got married, but I'm sure the symbol of the two dieties of the bonding partners show up.

Correct - the altar inside the Sanctum of the Twelve shows the symbol of your character and your partner's guardian deities during the ceremony (one one either side), but apart from that, the only other benefit your chosen guardian used to grant was a slight native elemental resistance to that particular element (for instance, my character's guardian, Azeyma, granted a slight resistance to fire), although that was abolished when Stormblood launched.

Back in the early 1.0 era you could also evoke your guardian during levequests in the form of a buff called 'Guardian's Blessing' which increased the EXP you gained, but that was replaced by rested EXP and Guardian's Blessing was removed.

[Haprimac]

They also affect the results in Valentione fortuneteller's sidequest from what I can tell.

[SturmChurro]

Would've been fun if we received a present in the mog mail when your 'birthday' arrives

For sure. I actually use my real birthday for mine, so it would feel good.

[Brightamethyst]

Would've been fun if we received a present in the mog mail when your 'birthday' arrives

Oh man. I never even thought about it before but I should totally start sending my friends presents on their characters' birthdays.