You may think that, 30 years after its release, you know every inch of Star Trek II: The Wrath of Khan*. And it's possible that you know this inch, but it's a really rare one that almost never gets brought up, and it's unclear if any of the footage still exists:
Khan originally had a child. In the movie the child first appears as Terrell and Chekov come upon the Botany Bay; Chekov sees the kid through a window briefly, and then the child scurries away. It's a moment of added tension as the two examine the wreckage of the ancient ship.
The baby doesn't show up again until the very end, and his return comes at a very sobering time. As the USS Reliant is in ruins, as Khan is all but defeated, he activates the Genesis Device, which still sits on the transporter pad. Attracted by the bright lights, the baby crawls towards the device... which then detonates, turning the Mutari Nebula into the Genesis Planet and killing everybody.
Those are the only moments featuring the kid; the only photographic evidence I can find of the child is above, from a 1982 issue of StarBlazer. The article is called "The Man Who Saved Star Trek," about director Nicholas Meyer**. It's a pretty good interview, where Meyer is fairly savage about Star Trek: The Motion Picture (saying he hired crew who worked on that movie because they would know how to NOT do it). Supposedly a shot of the kid in the window of the Botany Bay exists, but I haven't been able to find it.
The son shoudn't be confused with Joachim, Khan's right hand man. Many people often assume that he's Khan's son, but that isn't the case. In fact he's supposed to be a guy named Joaquin, who appeared in the original series episide Space Seed, which introduced Khan. A weird production glitch made it so the two characters have different names, and eventually it became enshrined by fans that they're different guys. Some authorized fiction has Joachim being the son of Joaquin, born on Ceti Alpha V, but growing up fast because of his superior genes (it's established that Wrath of Khan takes place 15 years after Space Seed). Side note: Judson Scott, who plays Joachim, is uncredited in the movie. That's because his agent was trying to get a bigger credit from Paramount but fucked it all up.
The son should also not be confused with this prank played on Ricardo Montalban:
This isn't the only deleted material from Wrath of Khan, some of which remains officially unavailable.
The film exists in three versions: the original cut, the director's cut and the ABC TV movie cut. The TV cut is strange, mostly filled with alternate takes. For example, the scene between Saavik and Kirk in the turbolift plays out in tight shots, and Kirstie Alley is more seductive. The director's cut includes more footage with Peter Preston, the young engineer who dies; the theatrical cut never establishes that this is Scotty's nephew, which makes his death more of a pay-off.
Then there are scenes that have just disappeared and are not even included as extras in the newest Blu release. There's a brief exchange between Kirk and Spock just after the Kobayashi Maru test where Spock reveals that Saavik is half Romulan; this information informs the fact that she later cries in the movie. Her half-Romulan heritage has become an accepted point of the expanded universe canon, but has never been mentioned in the official works. Also cut is a moment at the end that indicates Saavik totally boned Kirk's son David.
Thanks to Ray for sending over scans of the StarBlaze article!
* which means you may know the movie's official title is actually Star Trek: The Wrath of Khan.
** for spoiler lunatics in the audience: the February 1983 issue of Starlog has, on its cover, a blurb about an interview with "The Man Who Killed Spock." That would have been on stands six months after Wrath of Khan was released - your brains would have exploded back in those days.