Build a vast underground headquarters as humanity's last hope of survival. The game would be similar to Dungeon Keeper, Sim Evil, Evil Genius, Dwarf Fortress, etc. but would come complete with the following features:
Unrealistic structures you can build, such as moving buildings (some of which are
really just massive missile launchers or elevators for Evangelions), trains that carry cars, huge empty rooms with escalators that move slowly, holding rooms for Evangelions, etc.
The GUI would resemble the command center from the series, with the floating 3d display, Magi computers giving you advice, and lackeys giving you useless updates ("5000 meters! 4000 meters and closing! Energy Reaction Detected!") in japanese, with english subs, because the japanese voice actors are better.
You get to play as Shinji, Rei, or Asuka when an Angel attacks, and when playing on hard mode you suffer from random, crippling bouts of "spurting and burning pathos." If you're more into resource gathering and RTS in general, you can just default to the dummy plug every time. Works like a charm, except that you have to wash the blood off the buildings. . .
Speaking of which, you'd have to try to keep the whole quiet, so that the public doesn't catch on. Yes, it is stealing from Evil Genius and others, but this would be well executed, it is a different game, and it was an issue in the series. Moving on . . .
Deploy classic huge weaponry and tactics: Huge laser gun powered by all of Japan, N2 mines, aircraft carriers and high-tech helicopters, and a 1:1 scale balloon of Evangelion.
Perhaps the best feature would be the detailed, text-based output, the parts that are always there saying "left arm cranial servo damaged, sensory node 56 failed, synchro disconnect main 23," etc. but always with meaning, not random. So when the humans finally invade, you can say "oh, they've gotten past 26 special armor plates and into sector D, time to deploy the poorly armed guards in orange and fill the hallways with red goop that will solidify and buy some time, but will mostly just look cool." On this note, there WAS one game that did this, and I think it was called Microsoft Combat Flight Simulator, and when flak exploded right by you there was red text at the side that told you which aelerons were hit and whether the starboard coolant intake valve was damaged, cockpit pressure issues, and on and on.
But no. The BEST feature would be the confusing japanese-style provoking cut-scenes with those crazy philosophical messages. Yeeeaaah, lifted right from the series . . . the last two episodes and the endings on the movies. Brings back memories.