The Mother-Child Bond
Within the universe Shinji and Asuka both have "souls" or some meaningful part of their mother inside the Evangelion, presumably in the Core. This is pretty obvious in Shinji's case:
- Episode 1, Unit-01 moving by itself to protect Shinji, something Ritsuko immediately notes should have been impossible
- Episode 16, While trapped in the Sea of Dirac, Shinji manages to (whether he realizes completely or not) make contact with his mother right before the Eva goes bonkers and breaks out of the Angel
- Episode 21, Flashback: Yui, while at Gehirn, takes part in a contact experiment with a then partly constructed portion of an Eva and disappears inside it.
- At the end of End of Evangelion, Unit-01 is floating out into space as Yui talks about the loneliness of existing forever.
Among other little hints here and there. In Kyoko's case, Asuka's mother is less explicit but we see the same sort of things:
- Episode 22, Flashback: Kyoko undergoes a similar Contact Experiment as Yui, she doesn't get absorbed but's clearly crazy and commits suicide
- Near the beginning of End of Evangelion, Asuka is in Eva Unit-02 at the bottom of the lake, has a similar little discovery as Shinji from Episode 16, except this time Asuka is well aware that her mother was in the Eva
Toji and Rei: they're not so explicit. Rei is a special case because she's in some sense an Angel. With Toji, there's some possible indications that his mother, who has been dead since at least the beginning of the TV series, had some connection. The entire class had family that worked for Nerv.
Other than any sort of in-universe explanation, the idea of the mother-child bond actually came from Yoshiyuki Sadamoto, the author of the original Evangelion Manga. From the Deluxe version of "Der Mond" a Sadamoto artbook that contains a lot of Evangelion related work, there's an interview:
When the very first meeting was held before the title had even been decided, Anno had already provided the theme of "a battle between gods and humans". Both Anno and I -- our generation -- was influenced by Go Nagai, so making something on a grand scale meant it ended up like "Devilman". The character design request from Anno was that "the lead character is a girl, and has an older-sister type figure like Coach next to her," so it was structurally similar to "Gunbuster". So I first designed an Asuka-type girl as the lead character, but after "Gunbuster" and "Nadia" I felt some resistance to making the lead character a girl again. I mean a robot should be piloted by a trained person, and if that person just happens to be a girl then that is fine, but I couldn't see why a young girl would pilot a robot... So I remember saying to Anno, "It's a robot story, so let's make the lead character a boy." And just about that time, I was watching the NHK [public TV channel] program "Brain and Heart" and learned about the existence of the A10 nerve, and I told Anno about the idea that popped into my head at that time. That was the idea where "the dead mother is inside the robot, which is operated by mental/psychical bonding with the child. Moreover, parent-child relations are parched/strained due to the death of the mother at a young age." As soon as I had this idea I was filled with confidence that "This will work!" and I just whipped out a setting drawing. That setting drawing became the character chart for the Planning Papers.