Lucia is one of the orphans in Skyrim who can be adopted by the Dragonborn. To initiate this, she must be given a septim and asked why she is begging. She will explain that her mother died, and her aunt and uncle kicked her out when they took over her mother's farm. She then arrived in the streets of Whiterun City, where Brenuin, showed care for her, and told her to ask the townsfolk for septims. She can be adopted if the conditions for adoption (owning a home with a furnished child's bedroom) are met.
Lucia has the same name and a somewhat similar appearance to a child character in Fallout 3, another Bethesda game.
Once adopted, some of the things she may do when her new parent comes home include: ask for an allowance, present some flowers she picked or other gift, ask the Dragonborn to play with her, ask if there are any gifts for her or ask if she can keep a pet.
If she isn't the only child adopted, she will interact with the other child, playing or verbally fighting.
The farm from which Lucia was ejected cannot be found. It is possible, given that she is an Imperial, that it's in Cyrodiil.
Lucia says that Brenuin is the only person in Whiterun who has been kind to her, and that he advised her to take up begging, but Brenuin doesn't mention her.
After nightfall, but prior to when she goes to sleep, she can often be found in the Bannered Mare, sitting and mingling with the clientele as well as cleaning and doing other tasks as if she worked there (at some points it appears that she is actually drinking along with the adult patrons). Around 12:00-12:05 AM, she leaves the Bannered Mare and wanders to the back of the building, where she "goes to bed" on the ground by its southeast corner. All of this seems to suggest that she has a similar arrangement with the Bannered Mare as Blaise has with Katla's Farm and Alesan has with the Windpeak Inn.
If she is killed through mods, a courier will arrive and give a letter of inheritance to the Dragonborn along with 100 gold minus the usual Jarl's tax.
Lucia has remarks related to Whiterun which she will make from time to time. For instance, she will remark on playing with Lars Battle-Born, Braith and Mila Valentia, citing she doesn't want to play with Braith anymore because she's a bully to her and Lars.
360 When dialog is initiated with her in Lakeview Manor, it is possible to become locked in the conversation mode, where Lucia's dialogue does not occur, and it is impossible to exit.
360 If Lucia resists when she is told to do something, i.e. do her chores or play outside, and a female Dragonborn lets her get away with it, she may respond by calling them 'pa' instead of 'ma'.
360 After being given a new dress, Lucia may have no dress equipped the next time the Dragonborn returns home. After that, she will equip whatever dress given her, but will be in her underwear again the next time they return home.
PC When Lucia enters Breezehome she may get stuck near the fire. When approached or talked to, she only says "I can't wait".