Roman family names often reference obscure old Latin or Etruscan words.
The Roman gens by the name of Lucius, was originally called Luscius (LOO-ski-oos). And so the name comes from “luscus,” meaning “one-eyed” (with the added meanings “one eye squeezed shut” and “half-blind”).
Of course, the newer spelling and pronunciation allowed for a lot of flattering puns on Luciuses being born in the light, or showing up at first light, and so on.
Saint Lucia was either a member of the Lucian gens, or from the family of one of its freedmen. And since her mom was allegedly rich, and since her mom had a Greek (Syracusan) name but she bore the usual female gens name, she probably was a highborn Roman girl.
“Luscus” is related to “lux” and words about light, but also related to Greek words about glimmers and gloom. So the implication is more about “partially dark” than “partially open and full of light.”
There is a Latin word “lucius,” but it’s the name of a fish. (Which is why the Lucy family sometimes bore fish on their heraldry.)