Anno has stated that quite a few of the character's first names are little references to other works:
The first names of Misato and Kaji (Ryōji) is a reference to the characters Misato Izumi and Ryōji Sawada from Minako Narita's manga Aitsu.
The first name of Asuka's mother, Kyoko, is a reference to the character Kyoko Nakamura from Shinji Wada's manga Sukeban Deka.
Aoba's first name, Shigeru, is reference to and a pun upon Kihachi Okamoto's 1974 film Aoba Shigereru.
Toji's entire name (Toji Suzuhara), Kensuke's first name, and Hikari's last name (Horaki) are lifted from characters in writer and film-maker Ryū Murakami's novel Ai to Gensou no Fascism.
Shinji is a reminiscence of Akira Fudo from Devilman, another lithe, dark-haired teenager with a sensitive personality who is forced to adopt monstrous powers and to undergo personal development in order to save the world. Unsurprising considering that Devilmanwas a major influence for both Anno and Sadamoto.
Visually speaking, Sadamoto based Shinji's character design on Nadia from his previous work Nadia: The Secret of Blue Water. He described him as Nadia with shorter hair and without her earrings.
Though Rei is more often thought of as a character who is copied, she shares many similarities with Chirico Cuvie from Armored Trooper VOTOMS, most obviously their blue hair, cold, unemotional exteriors and as being vessels for Godlike figures, both of whom they ultimately destroy or defy.
In the Super Robot Wars games featuring Evangelion and Gundam (UC timeline), Misato gets a crush on ace pilot Amuro Ray, alluding to their seiyuus' other famous anime roles as Sailor Moon and Tuxedo Kamen respectively. She also makes some Sailor Moon comments when admiring the Nobel Gundam from Mobile Fighter G Gundam on Super Robot Wars MX, and points out similarities in voice with Vega from Gear Fighter Dendoh and Murrue Ramius from Mobile Suit Gundam SEED (two other characters voiced by Kotono Mitsuishi).
Officially, Kaji is "based" on Col. Paul Foster from the TV series UFO, although Gainax noted that Foster was not nearly the lech that Kaji appears to be. Some viewers have also suggested that Kaji has elements of Emma Peel from the British spy series The Avengers (1960s), as this chart by a Tumblr user shows.
Keel Lorenz is an allusion to the ethologist Konrad Lorenz. He was originally named Konrad, too. Konrad Lorenz developed the concept of imprinting in terms of mother/child relationships. For example, say you've got a duck's egg in an incubator, and it's ready to hatch. The eggshell cracks, and the chick's head pops out. It sees you, and thinks, "Mother!"
Yui Ikari is based on Nadia's mother in Nadia: The Secret of Blue Water. The long-dead mother of the protagonist who is mourned by their surviving husbands (both who are engaged in bloody armed conflicts and have grown distant from their children); both mothers whose souls reside close to their children, protecting them from harm and giving them glimpses of their past and purpose in life.
Similarly, Kyoko Zeppelin Soryu is Kushana's mother from Nausicaä of the Valley of the Wind, especially in how their insanity leads them to believe a doll is their child, while seeing their real child as an impostor. The case of Kyoko does, however, explore in greater depth and detail how such a madness in a parent would warp and damage their child's psyche.
Evangelions are also essentialy God Wariors from Nausicaa.
Toji and Kensuke are allusions to characters from Ryu Murakami's Ai to Gensou no Fascism (The Fascism of Love and Fantasy), from which Anno borrowed much of the psychological material.
There's too many references to various world religions (particularly Judaism and Christianity) to list, or possibly even detect. For a start, all the Angels are named after Angels from Jewish, Christian and Islamic theology and incorporate subtle parallels with their counterparts' names and functions.
The style of the opening is based on the classic sci-fi show UFO. Furthermore, the character design for Gendo Ikari and Kozo Fuyutsuki are animated look-alikes to the actors Ed Bishop and George Sewell as characters Ed Straker and Alec Freeman (the commander and second-in-command from S.H.A.D.O., respectively). Also the "blink-and-you'll-miss-them" title cards that appear at times are taken from the show.
There are multiple references to Michael Crichton's The Andromeda Strain, mostly the film version (the use of the "601" graphic to showcase the MAGI having a computer analysis error, the design of the Geofront resembles the Wildfire complex when seen from the side, and most importantly the episode "The Lilliputian Hitcher" is almost-but-not-quite a Whole Plot Reference).
Misato initially drinks "Yebisu" beer, a real brand that is depicted correctly down to the label, followed by "Yebichu" beer, a parody of the real label which references the manga Oruchuban Ebichu, of which both Anno and Japanese voice actress Kotono Mitsuishi were fans. Anno later developed it into an anime at Mitsuishi's recommendation.
A blink-and-you'll-miss-it one comes in form of one of Shinji's t-shirts, which has the logo of the British New Wave Music band XTC on the back. Seeing as quite a few of XTC's songs revolves around some of the same themes as Evangelion, the reference is likely more than just a superficial one.
The Human Instrumentality Project was named after Cordwainer Smith's Instrumentality of Mankind, though they are not otherwise related.
In turn, the Hedgehog's Dilemma is an allusion to Schopenhauer. It's also been theorized among fans that the original ending alludes to and borrows from some of the existentialists, such as Sartre and Kierkegaard.
Also of note is that the Working Title for the last episode as per the series proposal was "The Only Neat Thing to Do", the same title as a sci-fi short story written by Alice Sheldon under her Moustache de PlumePen Name of James Tiptree Jr.
At least some of the battle themes on the OST consciously evoke the Klingon theme from the original series Star Trek films.
The poster for the movie has a resemblance to the poster for an old film named Rosemary's Baby◊.
The title of the latter half of the film, "Magokoro wo, Kimi ni," comes from the Japanese title of the movie Charly, which was based on the short story/novel Flowers for Algernon.
The scene where the military attacks NERV HQ is an homage to Hideaki Anno's favorite film, Battle of Okinawa, right down to the use of flamethrowers.
Shinjis and Asukas Masochism Tango is disturbingly similar to that of George and Martha from Who's Afraid of Virginia Woolf?: Asuka is disgusted by Shinjis lack of assertion, Martha is frustrated with Georges, attempts by either at appeasement only aggravates them further, and both of their frustrations have to do with sexual performance and assertion—Shinji wont cross the "Wall of Jericho" and notoriously settles for masturbating by her bed when shes comatose while George is implied to be unable to perform, and both are Ambiguously Gay. Both Shinji and George ultimately both snap and lash out destructively, complete with an attempt to strangle Asuka/Martha.
One of the two unused, proposed endings was even closer to Devilman's ending than the final version.
The manga seems to have shout outs to Star Wars, of all things.
For example, on a NERV writeup of Toji, there's a documentation of his Midichlorian count.
In one chapter, Gendo also pulls a Darth Vader on Shinji, which is reminiscent of the fact he loses a hand in The End.
Misato mutters "I have a bad feeling about this" in the episode with Jet Alone.
Devilman and the Ultra Series get multiple shout outs throughout the series, some subtle, others overt. According to Anno the Devilman ones were unintentional and he only noticed the similarities when they where pointed out to him.
Many of the parallels between Ultraman and Eva are deftly explained in this article.Anno himself has been an avowed fan of the Ultra series since childhood, even being tapped to direct the upcoming Shin Ultraman feature film.