Accent Depundent: MIB agents are given code names taken from the alphabet. For some reason, however, the head agent, Z, is referred to as "Zed" rather than "Zee" which doesn't makes sense to those who are not aware of the UK/Canadian pronunciation of the letter.
The Cast Showoff: Tommy Lee Jones speaks fluent Spanish in real life, used in the opening scene when K questions all of the Mexicans attempting to cross the border (and trying to smoke out the one who's really an alien).
Enforced Method Acting: Tommy Lee Jones was not amused by the script he was given, so he ad-libbed a good deal of his dialogue. Look closely enough and you can see Will Smith trying to keep up, as expected of the fresh recruit. The resulting synergy is well over half the movie's charm.
Science Marches On: A minor one. When J heads to the morgue to retrieve Orion, he hilariously tries to convince Laurel that the cat is a witness in a murder investigation, unaware she's being held hostage by Edgar. Fast forward about ten years and advancements in forensic science mean that animals who 'witness' a murder can be checked for any forensic evidence the killer may have left behind.
Stunt Double: In one Agent K drives the car up the walls and onto the ceiling of a tunnel, and J, not wearing a seatbelt, flips overand in a rather infamous blooper, briefly gets a stunt-double-induced Race Lift.
Talking to Himself: For the bit when Edgar is taunting the Bug before it takes his body for a skinsuit, the Bug's lines are also supplied by Vincent D'Onofrio in a gravelly voice reminiscent of the one he later used to play Wilson Fisk in Daredevil (2015), and altered to sound electronic.
Technology Marches On: K shows J a small alien disc and says "They're going to replace CDs soon". At this point, it's doubtful. It was based on Sony's MiniDisc, which did indeed fail to be the wave of the future. Also, J's introduction is him chasing down an alien on foot to arrest him. These days, the fleeing suspect probably would have been tazed, ending the chase rather quickly.
The scene where Mikey gets killed at the beginning of the film was not originally planned. The creators added it in because they realized that most people would have sympathized with Mikey had he died as is, and wanted to ensure no sympathy was to be gained for him prior to being killed.
Will Smith's line "It just be rainin' black people in New York!" was improvised but so hilarious that they decided to leave it in.
In fact, a lot of the lines were this. Tommy Lee Jones was famously dissatisfied with the script and so he took it upon himself to "fix" it by ad-libbing a good deal of his dialogue and one-liners. The result was Will Smith, not to be outdone, having to play along too. It was hilarious enough that Barry Sonnenfeld just didn't care and threw it ALL in. And the movie ended up being praised for its witty dialogue, in case you needed more proof how awesome Tommy Lee Jones is.
The little alien in the mech body was originally written for earlier in the film. Originally, K originally introduced J to a bartender, who would lift his face up to reveal a body made of light. Then Phil Tippet had the idea to have his face open to reveal a pilot. Everyone liked the idea so much they moved the idea and made it plot-relevant.
Originally, there were going to be two alien ships looming over the Earth, demanding the return of the galaxy: the Arquillian ship, and a Baltian ship (Rosenberg was originally a Baltian in the script). Edgar came to Earth to deliberately start a war between the two races, said to be negotiating peace, so he can feed off the resulting war. The initial volley is compared to two gunslingers using a giant rock as cover. Eventually, they decided the Baltians were not necessary to the plot, so some work was done in post-production (dubbing the aliens' conversation into "alienspeak", for one), and long story short, Rosenberg became an Arquillian.
Originally, the Bug was going to be a puppet, but it wouldn't have been able to move around and fight. To have a more exciting final showdown, they spent 4.5 million dollars creating a CGI Bug.
The movie wasn't originally going to be set in New York, but Barry Sonnenfeld decided to change it because he believed that aliens would most likely be drawn to New York because nobody would notice their weirdness.