Director Shawn Levy told a National Public Radio interviewer that once the Smithsonian Institution agreed to lend their name to the film, their curators were very helpful and willing to share information about the real-life characters represented in the movie, but Levy had more trouble with the intellectual property-holders for the fictional characters represented. For instance, in the Darth Vader cameo, a LucasFilm employee came on set to observe the scene and tell the filmmakers what Vader would or wouldn't do.
Hank Azaria filmed test scenes (some of which are on the DVD) of his character Kahmunrah with various accents including American Southerner (from Alabama) and English Cockney, before settling on a Boris Karloff impersonation.
When first introduced to Napoleon and Al Capone, Ivan the Terrible says his title is mistranslated, and he calls himself Ivan the Awesome. Although obviously meant to be a comical line, it is not that far from the truth: Ivan's title is indeed a bad translation. A more accurate form of the original Russian term, Ivan Grozny, is "Ivan the Fearsome", "the Magnificent" or "The Admirable". It is important to emphasize that, in Russian, the word "grozny" does not have a bad connotation.
Octavius's line, "Stay alive! I will find you!" before jumping off the windowsill is a reference to Daniel Day-Lewis line in The Last of the Mohicans (1992) where Lewis's character says the same thing before jumping off a cliff.
In the original script for movie, Kahmunrah (Hank Azaria) revealed that he actually murdered his brother Ahkmenrah (Rami Malek) presumably for the throne. Which is also possibly the reason why Ahkmenrah is so young, and Kahmunrah looks older.
In the original script for Night at the Museum: Battle of the Smithsonian, Ahkmenrah (Rami Malek) had come to the Smithsonian with them and had a fight about the combination of the tablet with his brother, Kahmunrah (Hank Azaria) it is also revealed that in the end of the movie, it was originally Ahkmenrah who held a dagger to Kahmunrah's throat throwing him in the portal rather than Larry.
The motorcycle that Larry uses as a getaway vehicle is a rare 1942 Harley Davidson XA Military Side Valve 750cc Flat Twin with attached sidecar. This was Harley's copy of the German BMW/Zundapp military motorcycles.
The Smithsonian does host a giant squid exhibit, but this was forgone for a giant octopus in the film. For some reason, however, the film's marketing in Japan (where the difference between a squid and an octopus is common knowledge) consistently referred to the octopus as a "squid," even though it is never directly identified as either animal in the film.
The trivia item below may give away important plot points.
Near the end of the movie, Larry and Jed notice that Amelia Earhart has flown off in the wrong direction. Later, when Larry meets the human lookalike, she asks for help getting to the miniature room because she gets lost. This is a reference to Earhart's last known flight where she disappeared, not having any idea where she and her navigator were.