Me
nphina
I'm not sure what secrets are stored away at SEHQ, but to get an answer to this question in Eorzea, you'd probably need to ask an Ascian. Even the word "planet" is uttered very few times in the game, mentioned only by Ascians, Sharlayans, and (bizarrely) the common guards that explain what Aetherytes do (the gates were built by Sharlayans, though, so maybe there was some imparting of knowledge on those related to the project).
In a quote from the distant reaches of the ancient past (2010), "Though it is the name of the entire planet, the term is very rarely used by its inhabitants, for to understand the concept of a planet, one must first know of other worlds, and most on Hydaelyn do not."
For this reason, I wouldn't get your hopes up about deeper astrophysical exp
loreation.
In Eorzea in particular, considering the Sixth Astral Era was an intellectual dark age compared to the age of the Allag (Third Astral), you won't be like to hear much of this talk. The moon isn't just
named Menphina, it's oft thought to
be a physical manifestation of The Lover herself.
HOWEVER, even if Eorzeans might not know a name for the sun or moon aside from the names we give them relating to the Twelve, I can say this. Menphina is known as the Keeper of the Twin Moons, and we know that many consider the moon to be an incarnation of her. This would mean that, to Eorzeans, the sun would be an incarnation of Azeyma, who is our star's keeper.