Yuki Nagato is an obvious Rei Ayanami Expy. Not only due to her overall character, but also to her name, which is basically a pun to Rei's. "Rei" can mean "cold" while "Yuki" can mean "snow", and Nagato and Ayanami were both Japanese warships that served in World War II.
The plot of Disappearance can be seen as re-creating Rei's actions in End of Evangelion. Yuki connects with her own emotions and the people around her, and after a repeating cycle of suffering decides to rebel against her creator and use her full power to change the world; but gives the final decision to the boy who has been by her side all along. Very different plot, but similar character motivations.
Koizumi echoes Kaworu Nagisa from Neon Genesis Evangelion, actually more than Yuki to Rei Ayanami. Koizumi and Kaworu are both new transfer students with perpetual smiles, messy haircuts and hidden supernatural powers who belong to shadowy organizations, propose philosophical questions and extert homoerotic interactions with the main character. Later volumes only increase the similarities, as just like Nagisa is said to be not just an angel, but the incarnation of the most powerful of them, Koizumi's role is revealed to be not a mere member of his organization, but their founder and most powerful esper.
While explaining the concept of Moe to Kyon, Haruhi holds up two manga anthology magazines as examples. One of them has Asa and Kaede of SHUFFLE!.
The book Nagato gives Kyon is Hyperion. It's a book/series about time travel, aliens, bizarre powers and things that may or not be gods/God... Hm...
More than that: after reading Hyperion, one may gain a whole new level of insight into what Yuki is and what her (and the Data Overmind's) motivations are.
In the second novel, they are described (Disproved in 4th book, Rise of Endymion) as made up of 3 factions. The Stables, who want to continue their relationship with humans. The Volatiles, who just want to kill all the humans. And the Ultimates, who just want to build their god, the Ultimate Intelligence. Guess who represents the first 2 factions soon after...
The scene in which Nagato gets impaled through the chest may be a Hyperion reference too.
The book may also just be more of a thematic fit with the early episodes more than anything else, as it deals with a group of people from diverse backgrounds — each of whom has their own story to tell — drawn together on a pilgrimage to face a mysterious, almost supernatural force.
The fight with the cave cricket references the arcade game Mushiking; when the cave cricket is healed partway through the fight, we get a green beetle flying over the cave cricket releasing sparkles everywhere and the line: "Scarab healed your wounds!"
In "Mysterique Sign", when Yuki exposits about the alien's arrival, the montage begins with an homage to the opening shot of The Thing (1982).
In the episode "Remote Island Syndrome Part I", Kyon references Infant Island from Mothra and related films, and Suehiro Maruo's manga The Strange Tale of Panorama Island.
In "Remote Island Syndrome Part II", the 'culprits' of the murder appear as black shadowed figures with slanted yellow eyes. This is also how unidentified criminals are portrayed in both the anime and manga of Detective Conan.
Anyone who has watched Detective Conan will instantly spot how each new character important to the mystery gets on-screen text stating their name, occupation, and age.
Haruhi imitates the sprites from the video game Ace Attorney, as well as the dramatic close-ups and, but of course, the "OBJECTION!". Itsuki runs with it and imitates Edgeworth.
Mikuru also gets in on the act, bowing her head with her hands together just like Maya Fey.
Kyon seems to go with it, too: he's evidently playing the Mia to Haruhi's Phoenix, since he's mostly shown during the explanation with his arms crossed, which is one of Mia's default sprites (especially when channelled by Pearls), wears his collar open like Maya's and Pearls' costumes go when channelling Mia, and was the actual one who figured everything out, if his narration is anything to go on there. Like Mia, he also gets to add a little bit of the explanation to the reveal.
The aforementioned episode is also a shout out to Star Blazers, which also took place on giant spaceships with giant cannons. The leader of the Computer club even quotes the Catchphrase of Star Blazer's season 1 episodic villains by saying: "All hail the computer club!!!". The original line said by the villains every time they lost to the protagonists' crew being: "All hail the Gamilon Empire!!!"
In "Someday in the Rain", Kyon and Koizumi are playing a real TCG called Dragon All-Stars that features various anime/light novel characters; among the cards shown are "Lina and Naga" and a Gourry card.
Kyon notices that Time Traveller Mikuru has never been inside a fashion store — rapidly changing clothes on mannequins in the window of a fashion store are part of the Time Passes Montage made famous by HG Wells' The Time Machine.
Also in Endless Eight, after being told about the endless recursion of time, Kyon utters something similar to Goemon's catchphrase from Lupin III. Also at one point, after Koizumi tells him to say "I love you" to Haruhi, he replies "Ore no turn!"
Yet again in Endless Eight, during the festival, Yuki buys herself an Ultraman mask.
Each time Kyon wearily removes the frog costume during the "Endless Eight", he makes reference to a different anime/manga. For example: "Now I know how Unit 00 felt after being hit by that beam."
For extra fun, remember who piloted Unit 00. Then look at Yuki.
In Endless Eight part four, he references Mobile Suit Gundam 00: "Now I know how it feels for Virtue to do a purge".
And he also goes 'gerogeeroo' at the end of this scene in at least two of the loops. After Haruhi and Keroro have both been referenced in Lucky Star, this completes the triangle somewhat.
In the anime adaptation of Sigh Kyon mentions Apathy Syndrome, a reference to Persona 3.
Also in the Sigh anime: "Target in the center, then press Record."
Again in Sigh, Kyon remarks that a waitress outfit is unsuitable for a battle scene. One of the alternatives he suggests is a plugsuit.
When Haruhi starts dancing at first base, Kyon wonders if she's trying to reduce the magic points of the pitcher; the move "Odd Dance" does that in Dragon Quest.
Also in the Sigh anime: Kyon tells Koizumi to cast "Thwack" on himself and die.
In Dissociation, Yuki is reading Godel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter. Yuki could start the trope on Suspiciously Apropos Literature.
Another Wells reference is made by Koizumi in the 9th novel, he says that Nagato can't catch a human virus, as she is a real alien, and not a "martian from the past."
The name "John Smith" in itself is a direct reference to the British series Doctor Who, which is all about (you guessed it) time travel, sliding through alternate worlds and whatnot. Quite fitting for as alias for Kyon when it served as the alias for the Doctor as well.
Actually, John Smith is an extremely common alias used, just like John Doe is.
According to Kyon, Taniguchi and Kunikida are less harmless than a healer slime showing up alone.
In The Melancholy of Asahina Mikuru, from the 6th novel, Kyon referenced Veritaserum.
Compare Kyon's slow-mo fall through the window in Disappearance to Spike's in Cowboy Bebop.
Dub and translation
In each English book cover of the series, there is a little symbol on the top normally relating to what's going on in the book. In the sixth novel, the Wavering of Haruhi Suzumiya, the symbol just so happens to be the "Groundhog Day" Loop 's page image.
At the beginning of "Live Alive", Kyon gets waylaid by two people dressed as Hard Gay and Akihiro Miwa.
A fairly unknown reference in the Western fanbase, an early episode has Kyon almost incomprehensibly slurring a remark that roughly translates to, 'This has got to be a lie.' The reference is to an internet meme originating from Kamen Rider Blade where the main protagonist's slurred Japanese is referred to as 'Ondouru-Go' by the Japanese fanbase. In fact, Kyon's quote is copied exactly word for word from one of the more popular 'Ondouru-Go' phrases.
In the English dub of Melancholy (episode 3), during the second part of Yuki's initial exposition speech at her apartment about Haruhi and her being an alien an all that, you get this sequence:
Yuki: There is a reason she choose you rather than someone else.
The same shout out (no way/way) appears in chapter 30 of Dan Simmons' novel Endymion, between Endymion and the "ship" AI. Endymion is a sequel to Hyperion, which Yuki had just read and given to Kyon. It is reasonable to suppose Yuki was reading the sequel at the time and got the response there. There are neat parallels. Endymion/Kyon is protecting a girl (Aenea/Haruhi) with the fate of the world in her hands, and ( ship AI / Yuki ) are trying to protect both.