I managed to grind out something resembling an outline.
Premise:
Gendo Ikari is a brilliant scientist who takes his son, Shinji, on crazy adventures.
Characters:
Gendo Ikari:
Gendo Ikari (nee Rokubungi) is a brilliant, if unorthodox scientist. He’s created many weird and wonderful devices, any number of which are capable of changing the world for the better.
He’s also completely amoral and thinks nothing of interrupting his son’s schooling to further his agenda.
Unfortunately for everyone, it is an agenda that is as fluid as liquid helium and as mysterious as dark matter.
Yui Ikari:
Yui Ikari, the late wife of Gendo and mother of Shinji. A woman of great beauty and sufficient intelligence to attract a terrible human being like Gendo Rokubungi. Yui Ikari was attracted to the idea of a glorious future courtesy of Gendo’s gadgets, with her looking to market some of his less troublesome ideas.
Unfortunately for the Ikari family unit, Yui was starting to get bored with Gendo and his apparent inability to create a functional, marketable product and she divorced him and took Shinji away.
After Yui’s death, Gendo is forced to look after Shinji, though he doesn’t seem to mind too much after a few discoveries during the intervening years.
Shinji Ikari:
Shinji Ikari, the only child of Gendo and Yui Ikari. A regular 14 year old boy living in Tokyo-3. He’s not the brightest, bravest, most athletic or even most cunning. Average is the word.
Compared to his father, Shinji Ikari is a bit dull-witted. But then, Gendo is possibly the smartest man on the planet.
But Shinji is a good, kind-hearted young man, whose quick thinking has saved him and his father on numerous occasions.
Naoko Akagi:
Naoko Akagi is the brilliant, but somewhat unhinged, rival of Gendo Ikari. She, and her associates, managed to cause the Second Impact, an apocalyptic event that changed the world forever.
The super-science community had chided her for incompetence, but Gendo had shrugged it off as an accident with a comment about “eggs and omelettes,†albeit in a tone that came across as sarcastic.
After a few years of trying to fix things, Naoko decided that Gendo’s slightly sarcastic comment was right and she refocused her efforts onto the goals of the SEELE Committee.
Gendo always seemed to get in the way and cause trouble for them, but it was not always through malice or general unpleasant behaviour. There were occasions when their agendas coincided and he would help them out.
Usually, this involved Shinji Ikari, and when it became clear that such a thing wasn’t a coincidence, a plan was formulated. A plan that Gendo himself would be proud of: use of a family member.
Ritsuko Akagi:
Ritsuko Akagi was a few years older than Shinji Ikari, but she still had a rebellious streak in her. She was bright, attractive, but resented her mother for her absence as a parental figure.
Ritsuko had often wondered if she was an unwanted child, especially since her father had long since fled the scene. Her grandparents had always treated her with respect and love, even if her mother had been a little resentful for the career interruption the pregnancy caused.
Still, it was weird having your mother try to push you towards a young man for the sake of unclear scientific goals.
Asuka Langley Sohryu:
Asuka Langley Sohryu is a classmate of Shinji’s. She’s also something of a school idol. She’s rich, smart, beautiful, but something of a bitch to people who annoy her.
Unfortunately, a number of people annoy her, including her mother. For reasons that no-one can adequately explain, Asuka’s mother, Kyoko Zeppelin Sohryu keeps trying to get her to get close to Shinji Ikari.
Asuka barely recognised Shinji at first, but certainly became aware of some of the weird things that seemed to happen around him. Things like a man claiming to be his father dragging him out of class for some vaguely described task.
But she did also acknowledge that Shinji Ikari was not as dull-witted as some of his classmates.
Kyoko Zeppelin Sohryu:
Kyoko Zeppelin Sohryu is Asuka’s mother and a brilliant scientist, one that could rival Naoko Akagi in a purely academic contest, though she is less ruthless than her boss.
Kyoko works for Naoko at SEELE and is one of the head researchers, though something of an absentee mother, leaving her filthy rich but mentally inferior (Naoko’s words, not Kyoko’s) husband to raise Asuka.
Kyoko would love to make one of the breakthroughs that Gendo seems to make during one of his bathroom breaks, so she agrees to Naoko’s plan to get Asuka and Shinji together for the possibility that he might give her something valuable.