A single comprehensive article about Japanese culture would, of course, be an unworkable ordeal even for a Wiki devoted to that subject. In this instance, the focus is on instances in the manga and anime that are important and show off that culture's differing approaches on many matters; while opinion will inevitably leak through, let future editors keep this article professional and allow for counter-arguments rather than deletions unless the statement is of a fundamentally offensive nature.
Concern for family privacy
A common element to both the manga and the anime is an early scene wherein Kouta and Yuka decide to take the naked horned girl, who we know to be Lucy, home rather than call the police. At first glance, this seems merely to drive the plot, since turning Lucy over would lead to so much of what we saw not happening. Rather than a mere gift to the plot, this incident changes in the light of the culture surrounding them. The two friends/cousins know nothing of who this girl is. Putting aside her real identity, she could have been a high school girl who went on a boat with friends and got drunk, the horns the result of a prank. If this or anything like it had been so, and Lucy had a real family, it would have brought them some embarrassment for having a daughter in such a state. Having it known to the public would only make it worse. While nothing in the series confirms this hypothesis, it is entirely possible that Kouta and Yuka were hoping to get this girl to her family quietly and without worries of others learning their business.
Update: It seems that many Japanese fans also view taking the then-unknown girl home as a plot oddity as well. However, this does not negate the possibility of the above ideas having been on the mangaka's mind.
School and Location
Kamakura's town University is the rough equivalent of a 'community college' in the United States, a so-called 'safety school' for those students who still want a college degree but lack the skills and motivation for one of Japan's top universities. Kouta, perhaps owing to the enormous disruption in his life caused by his family's murder, finds himself attending this school despite signs earlier in life that he could do better. Yuka's choice to forgo a better school for Kouta's sake is an enormous indicator of her affection for him since she is likely sacrificing a future job with real earnings potential and respect.
Despite her status as the series' first victim and near-memetic reputation as a complete klutz, Kurama's secretary Kisaragi was a graduate of Tokyo University, called the Harvard of Japan. Entry into this elite school was a huge plot focus of one of Elfen Lied's possible inspirations, Ken Akkamatsu's Love Hina, the series whose character, Mutsumi Otohime, is possibly a direct inspiration for Kisaragi.
Mayu's transfer to a new school is something that happens all the time in Japan. Students often seek the advantages of the reputation of this or that school in a discipline of their choice and staying with relatives or friends nearby the better school is quite common, perhaps more so in Japan than any other modern country on Earth. What renders Mayu's circumstance uncommon, beyond the sad circumstances of her original home life, was the speed with which Mayu's mother, in essence, surrendered custody of her daughter to total strangers. Usually, even if the people a child is sent to live with are not directly known to the parents, they are known to close friends or family members, well enough that these third parties may vouch for their character. Mayu's mother sought no such verification, which sadly ended up telling Kouta and Yuka quite a bit about the quiet girl's ugly past.
Senpai and Kohai
Broadly defined, this is a mentoring system between two people, one slightly older than the other, but the devotion between the two often exceeds even familial connections, and while a great deal more informal than other such relationships, is often one of the most enduring and reliable. It is a relationship not unknown in Western culture; the 'batmen' who served British officers around the world and during the First World War are probably the best example. These junior officers were often known to lay down their lives out of personal affection and respect, with the senior officers ensuring that their former batmen were cared for all their days. These batmen and their service were the direct inspiration for the relationship between Frodo and Samwise in JRR Tolkien's The Lord Of The Rings. So it is for the people in such a relationship in Japan, only the service is after a time mutual and never ceases. It is worth noting that Love Hina's Shinobu never once referred to her crush Keitaro as anything but 'Senpai.' The two words mean Senior and Junior, respectively, but like many words in many cultures, their literal meaning is dwarfed by their definition as practiced.
At some point during her high school years, Yuka becomes responsible for fellow younger student Nozomi. This relationship might have started out as nothing more than a simple request by a teacher that Yuka watch out for the nervous younger girl, or Yuka, perhaps seeking responsibilities that took her mind off of the long-absent Kouta, took this upon herself. Yuka's residence at Maple House became in effect a place where she could defy her father's wishes, an outlet that gave new strength. Inspired by Yuka and her other friends Nozomi began to start to repair her relationship with her father, strained by years of his excessive efforts to protect her from her talents after the loss of her mother.
Office Lady and Boss
The relationship between Kurama and Kisaragi is similar to the Senpai/Kohai dynamic, though it is more in line with the connection of an Office Lady and her boss (although typical OLs work under several men and not just one). Their relationship is amiable and devoted, though ultimately platonic and based on mutual support. Kurama was not the ideal boss and Kisaragi was a far cry from the ideal assistant, but Kisaragi did her best to handle secretarial work beneficial to Kurama and he was likewise kind to her. Her death, though it occurs at the beginning of the series, is the next to last step in Kurama's unmooring from the Kakuzawas, and her loyalty and affection is remembered, as her picture is kept next to that of his own wife and Nana's. She is one of four women in his life willing to look past the hard exterior he puts forward, as all but Nana are taken from him by the Diclonius situation. Kisaragi was again not the best secretary, but for him, she wished to be, and that was all he asked of her. Sadly, it is an unneeded comment to the newly captive Lucy that likely seals her fate even as the series begins.
Shirakawa attempts to fill Kisaragi's role, but her mixed feelings plus her secret mission perhaps make this difficult. Her feelings were also more romantic, which, while possible between an office lady and her boss, romantic relationships between OLs and their bosses tend to be joked about rather than considered serious affairs.
Japan is rightfully able to boast of one of the world's lowest rates of unadopted orphans. Immediate family members are very quick to take in unfortunate minor children, but in cases where no family member is willing to take the child in, the child can be stuck in a primal Catch-22 limbo, frustrating would-be adoptive parents and eating away at the children's morale and sense of optimism.
A child without any close or known relatives might be asked to take the name of the family who has taken them in, but if any member within certain degrees of relation objects to this, then (at least as the law stands in 2012) the child in effect cannot be adopted. The family member in this case, even if they have no intention of caring for or otherwise being responsible for the minor child, is in effect given a veto over their destiny. It can be assumed that many of the children in the orphanage fall into this category, explaining partially what Lucy called their need to find someone even more miserable than themselves.
A frequent catching point for American readers and viewers is the relationship between Kouta and Yuka. Cousin marriage in Japan is only 4 in 1000 couples, as compare 1 in 1000 for the United States, where even the restriction on first cousins is not universal. It is entirely legal, and not at all frowned upon, though it is not as common as it was during times past. First Cousins are considered the legal limit--any relationship with a relative closer than this is considered both illegal and immoral. So long as no constant pattern of first cousin marriages develops in their descendants, the chances of their daughter having any health problems is four to six percent, as opposed to two to three percent for a child of a non-cousin marriage. With the loss of Kanae, it is now impossible for Little Nyuu to ever have a first cousin. It is possibly most useful to think of this aspect of their relationship as being how they first met and knew each other as children. In-series and in the real Japan, that is really all it is. Perhaps, it can even be speculated, the fact that cousin marriage was once far more common is used by the author to emphasize how mundane and traditional their relationship is, as opposed to that between Kouta and Lucy.
A sad historical look at a figure whose family took this too far can be found here.
A snarky and yet thorough look can be found here.
Many aspects of Japanese mythology are at play throughout the story of Elfen Lied, although many of these things have parallels in other cultures as well.
On a modern level, Kouta showed kindness to an outcast girl and developed a friendship with her. Misunderstandings within that friendship led to a terrible tragedy. On a mythological level, Kouta had interactions with a non-Human he did not know, and in his ignorance of the rules of dealing with her, he lost his family. From a modern POV, Kouta's only error was the innocent lie he told to attempt to spare Lucy's feelings. From a mythological one, his error lay in venturing outside what he knew, an action that cost him horribly.
Kanae is offered up the role of someone who has a terrible knowledge that no one will listen to, like the Trojan noblewoman Cassandra, among many others. Even though she was correct in what she was saying, she committed the sin of contradicting her elders from a mythological viewpoint.
While Yuka may have expected a much smoother path to Kouta's heart, her trials and tribulations and patience (relatively speaking) while he resolved his relationship with Lucy fits in with many a young woman waiting for her man. Even the instances where she is driven to distraction and off-putting behavior by her love for him fits in well many a young woman of legend.
Lucy is hardly the only protagonist, in or out of Japanese myth, given to rage-filled and even homicidal moments. Many a warrior and champion end up with great wrongs they must make right or live down. The concepts of Bushido in Japan and Chivalry in Europe were the result of samurai and knights commissioned for war still killing when the war was done, and taking what they needed from the weak innocents around them. Many legends concerning these violent former warriors see them encountering true kindness and love and then trying to turn away from the life they knew, with varying degrees of success. Many times, echoes of their past, like Chief Kakuzawa for Lucy make their newfound peace impossible. The story (also told in anime and manga) of Rurouni Kenshin, the fictionalized tale of a real-life samurai who drew much blood during the wars caused by the Mejii Restoration's reforms, is another such tale, though many in both legend and real life precede and follow it. In European cultures, such stories stretch back beyond Heracles and on through past the gunslingers of the Old West, perhaps iconically embodied by Alan Ladd in the film Shane.
One of the many contention points in the series is the scene at its end where Kouta loudly swears that he will never forgive Lucy's murder of his family when they were children. Kouta may seem unduly harsh to some fans, even if his vengeful words ring hollow in light of his subsequent sacrifice to save her.
It should first be pointed out : Kouta's rage is fresh and raw at this time; the profound revelation of who Nyu really is is a few days to a few weeks old, and he is overwhelmed. In-series, this is the card the reader sympathetic to Lucy is dealt.
Next to consider are the lives of his family, taken by young Lucy. No matter the tradition involved, for Kouta to forgive her, he must put their memories aside entirely to grant her forgiveness, and in Japan, this is perhaps ten times as difficult. For Kouta, granting Lucy peace meant literally dishonoring Kanae and their father.
Also keep in mind Kouta knows nothing of the orphanage, the puppy, or the role his innocent lie played in events. It is a cruel truth, but it is the truth, as is the fact that Kouta has no way of verifying her tale of an inner voice whose existence as a separate entity is itself debatable.
Finally, whether Kouta never forgave her or did so after the series, the most important thing to Lucy was that she was able to apologize for her actions. She would have wanted Kouta to accept her apology, but living long enough to give it was her primary goal--a goal she achieved. Kouta's love for Nyuu and the love he has for Lucy, or rather, the kind person she used to be, is a separate thing from his forgiveness.