l'll try explainig a few things about Elfen Lied.
Nyu is based on Chobits Chii, she also keeps repeating one word for a long period of time until she learns to speak normaly (if that was ever possible in her case

)
your refference to "New" is also in place, l guess it's time for a spoiler.
since lucy finaly escaped that facility she wanted to start a new life indeed, most probably away from everyone she ever met, but since she pissed off a whole bunch of "fate fairies" ( heheh )they placed her in a very bad position on the beach.
of all people should would ever want to meet, her oldest and dearest friend was just on the beach with his friend, now , lucy is a strong girl but she has also hilariously good memory, she instantly recognised the voice of her dearest friend that she betrayed ( boys voice doesn't really change that much with age) therefore she instantly created a blank personality, thats how nyu was born.
During the series you can clearly notice that Lucy and Nyu are conversing and that Lucy is teaching some things to Nyu, altough those are both the same person, Nyu seems to be very independent as an entity, and tries very hard to make Lucy feel better, at least to her own inept ability with a mentality of a child, or at least someone very innocent, and tries to make Lucy come with good terms with her past towards the boy, whos name l have forgotten, but shes as stubborn as ever, that series ending made me a bit sad, but l have to admit it was one of teh best series l saw at that time, made a huge impression on me
anyways, l reccommend this anime to everyone, great art, great story.