These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Draco in Leather Pants: HAL in a weird way; there are fans who insist he only attacked the crew out of self-preservation.
Though to be fair it was revealed in 2010 that in a way what happened wasn't HAL's fault. He was ordered to keep the true nature of the mission a secret, which conflicted with his basic programing, which caused his actions in the film. The man who planned the Discovery mission was LIVID when he found out what had been done to HAL.
Freud Was Right: Long phallic space-ships, docking bays opening up to receive them, music climaxing as the ship touches down, narrow slit windows with red light shining through, women emerging from doorways that look like vaginas, the Star Child. Really, this list could go on forever.
When Kubrick adapted the book to movie form, he changed a setting: instead of having Discovery head to Saturn and its moon Iapetus, he moved it to Jupiter and its moon Io. He did it because he couldn't create the special effects to make Saturn. Lo and behold, in 1979, the Voyager probes discovered that the next moon out around Jupiter, Europa, is very icy, and later observations have found it likely has a tidally-heated subsurface ocean of liquid water. Not only did it inspire Two Thousand Ten The Year We Make Contact, but today Europa is considered more likely to harbor extraterrestrial life than Mars!
Also, Pan Am, in 1968, was all but ubiquitous—it was the international airline for the US, and a cultural icon. Pan Am folded in 1991, partially absorbed by United Airlines; its nearest rival for "official airline of the United States", TWA, was bought in all but name by American Airlines in 2001. Obviously, Kubrick had no way of knowing any of this in 1968, and so naturally extended current tendencies in the airline world to space...but that doesn't keep the presence of Pan Am spacecraft from being hilarious Zeerust to modern audiences.
Apes, monkeys or general monkeying around... to Thus Spoke Zarathustra.
Older Than They Think: Most people who watch the film and do not know its age believe it to have come in the wake of Star Wars or thereabout - i.e., the late 1970s. Part of this is the impeccably accurate portrayal of modern spaceflight, technology, et al, and part because of the gorgeous quality of the cinematography and special effects, which rival Star Wars and make it appear as though it were made in the late 70s.
It's really hilarious to see people's reactions when you tell them that it was released before the lunar landing.
Seinfeld Is Unfunny: Some viewers may find that the "Star Gate" sequence reminds them of the audio visualizers in MP3 player software.
Shout-Out: Possibly. Move each letter in HAL's name up one letter, and you get IBM. This gets a brief discussion in the novelization of 2010.
Hell, even the computers look better than most of what came between this and the CGI era, or even the real life computers from The Eighties.
And the technique used to create the "Beyond the Infinite" sequence — a camera trick known as "slit-scan" — was impressive enough to be reused well into the early CGI era. It was later used for ABC's "This is the place to be" ads of the early 1970s, as well as the Whooshing Credits for Superman: The Movie (which improved on 2001's techniques by using a computer-controlled camera) and a whole bunch of pre-CGI motion graphics work in The Eighties.
Weird Al Effect: As time progresses, it becomes more likely that the first time somebody will see something related to the film will be as a Shout-Out made in another more current work rather than in the movie itself.