While I would be shocked if Gamefreak ever confirmed or even thought of this theory, I fundamentally believe that Lucy is Kieran and Carmine’s mother.
Evidence:
Lucy’s hair is black with a little red, not unlike Carmine’s. Purple, the other color besides black featured in Kieran’s hair, also appears prominently in Lucy’s design. All three characters also feature a yellow band of some kind.
Lucy and Carmine share similar body types, with both being relatively tall and thin.
Lucy uses almost exclusively serpentine Pokémon. The siblings both have almost serpentine eyes, and their storylines involve a number of snake-adjacent Pokémon, including Kieran’s Dipplin and Carmine’s multi battle against a Milotic, a Pokémon actively used by Lucy.
Lucy’s main gimmick is luck. Both siblings mention luck multiple times, including Carmine informing the player character that they’re lucky she’s there when they battle the aforementioned Milotic.
I’m really fond of this theory/headcanon/au, so hopefully I’ll find a way to do more with it in the future.
Update:
In Mochi Mayhem, you battle Kieran and Carmine’s grandparents, Yukito and Hideko, who would hypothetically be Lucy’s parents (so the siblings’ maternal grandparents) given Yukito’s “snake eyes,” similar to the siblings. In this battle, Yukito uses an Arbok, another snake pokemon.
