–Yuka, offering somewhat misleading instructions in an already confusing situation
In the opening chapters and first episodes of the series in both versions, three characters meet, two for the first time directly, and two who have met before but no longer recall it. As with the start of any narrative, the choices made here affect the entirety of the story, but at the same time, so much more is at play than the characters, and even the audience, could realize.
During Lucy's bloody escape from the Diclonius Research Institute, she received a head wound that pushed her off the cliffsides surrounding the island and into the ocean off Kamakura, Japan, washing up on Yuigahama Beach. At about the same time, a young man named Kouta arrived in town by train, there to attend a local college. He is greeted there by his cousin and childhood friend, Yuka, who once shared and still holds a romantic crush on Kouta. Sensing that Kouta is perhaps not recalling their times together as children, Yuka guides them to Yuigahama, where the sight of the naked, bleeding Lucy greets them as she emerges from the surf. After having Kouta give his shirt to cover the girl, Yuka suggests that they bring the girl, who seemingly cannot speak, back to the house provided for Kouta to stay at by Yuka's mother.
A rare convergence of the show's harshest critics and its most devoted fans often mock and criticize the seeming absurdity of this scene. It is to all appearances there only because turning Lucy over to the police would either end the story or move it radically away from what the author intended. But whether the mangaka intended it or not, much more can be discerned by a detached overview.
The first thing to consider is that Yuka is the only functioning adult in this sequence, or at least the closest thing to it. Therefore, unless her suggestion is something gravely untoward, it is likely to be followed. Lucy/Nyu is an infant in mindset at this moment, with her still-bleeding wound probably magnifying this. Kouta has the doubled burden of his amnesia and his post-traumatic confusion added to the bare possibility that he recognizes the strange girl they found but is fighting ever harder to suppress those painful memories. The other two are in fact so addled by their wounds, seen and unseen; it is even possible that they would still follow a genuinely strange suggestion from Yuka.
Another factor in this otherwise plot-driven choice is Yuka's desire for the fairy tale/romantic manga reunion with Kouta. While culture may be in play as well, Yuka could see a trip to the police station, the attendant questions and time spent there as further diluting what was already developing into a reunion not merely less than desired, but a downright awkward one. By getting this girl to the house, she would shake off her confusion, give them a phone number to call, and that would be that.
Events immediately after this would render all such reasoning moot. In short order, Lucy/Nyu's memory loss would be shown to be both long-term and intermittent. Questions would end up raised about the true loyalties of the Kamakura police. Yuka's plans for school and living would change, and she would gain a limited if a frustrating understanding of Kouta's condition. Within those same immediately subsequent events lay the seeds of others, including the fates of two characters destined to join the awkward trio in their new home. By meeting, reuniting and then quickly returning to Yuigahama, the involvement of Mayu, Bando, and Nana is assured. By this reasoning, a plot point that can seem eye-rolling instead becomes a spur-of-the-moment choice with its logic yet also quickly overwhelmed by the events begun and resumed at that spot. A sometimes-repentant murderer ends up in the care of the one she loves the most, who is also the one she hurt the most, overseen by a girl who unwittingly spurred one of her rampages, and who she once casually swore to kill. The story had only begun, but in this one scene, all the rest of it was stored up.