There's plenty of mysteries that unfold over the course of Fire Emblem: Three Houses. Byleth's heritage, Rhea's true identity, and the existence of the "One's Who Slither Below" are just a few of the riddles we get answers to over the course of the game. We never get any information about Byleth's mother, though. Be warned, there are spoilers for the main story of Three Houses!
Throughout the entire game, the only parental figure Byleth ever knows is Jeralt, their mercenary father. Jeralt is a man of relatively few words, and he doesn't have the best social skills, so you can easily see why the difficult topic of Byleth's mother is never properly broached by him. Instead, he dies at the hands of the villainous Kronya nearly halfway through Three Houses, never having discussed Byleth's bloodline with his own child.
Over the course of Three Houses, we come to learn that Byleth carries the goddess Sothis within them as a result of Archbishop Rhea attempting to reawaken the goddess. In order to try and reawaken Sothis, Rhea imbues the spirit of the goddess within the infant Byleth, thus causing the death of Byleth's mother. It's a brutal twist.
But other than that, Fire Emblem: Three Houses does little to shed additional light on Byleth's mother—until now. With the launch of the Cindered Shadows DLC, Byleth, Claude, Edelgard, Dimitri, and others take a trip into the sewers of the Garreg Mach Monastery, where they discover not Ninja Turtles, but a troupe of brand new students to befriend and fight alongside.
It transpires that one of the newcomers, Aelfric, actually knew Byleth's mother when she was still alive. Aelfric acts as a steward of sorts to the rabble living in the sewers, and Byleth can call on him anytime they wish to know more about their mother. Aelfric actually discusses Byleth's mother pretty openly, a sharp contrast to the rest of the game, wherein Byleth's past is mostly revealed in riddles and drips of information.
In an important touch, Aelfric reveals Byleth's mother's name: Sitri. At long last, we finally have a name to go on for Byleth's mysterious mother, who Aelfric tells us was absolutely infatuated with Jeralt as a heroic mercenary for hire.
Much like Byleth, Aelfric reveals that it was "very hard" for Sitri to express any sort of emotion. This is notable because Byleth is noticeably lacking in any sort of emotion throughout the majority of Three Houses, which Rhea eventually reveals is due to her merging the spirit of Sothis within Byleth.
These are the first concrete details we've ever had on Byleth's enigmatic mom. It's nice to finally have some background on the character that mothered Byleth, and the woman that Rhea would ultimately sacrifice for her own gains.
To cap it all off, Byleth's mom is actually buried in the grounds of Garreg Mach Monastery (a pretty cruel twist, considering its ruler killed her). If you visit it during any of the three main campaigns of Three Houses, you won't find a name. But with Cindered Shadows, the name "Sitri" can now be found on the gravestone in the Monastery. Hey, the more you know.
This all goes a long way toward helping Byleth feel like a real character with a real arc and not just an avatar. In Three Houses, it was easy to feel like Byleth only existed to connect you with the in-game world, not unlike Link in The Legend of Zelda. With Sitri finally coming into her own as an actual established character in the Three Houses world, Byleth has an emotional connection to someone apart from Jeralt, and there's been some light shed on one of the most elusive characters in the game.