I was reading through some of the entries concerning the Borg in the Star Trek Encyclopedia, and came upon a comment they had about Wolf 359 - it's the name of an actual star in space, it makes up part of the Constellation Leo. It's also the site of the first major fleet battle between StarFleet and the Borg. Take your mind back to the scene where Zef and Lily first walk out of that bar, and Lily sees a speck of light that is actually the Borg Sphere, and asks Zef what it is. He replies "That, my dear, is the Constellation Leo". Now, obviously he didn't see what she was pointing out the first time around, but we could probably assume that from their point of view the sphere was in the general area that the constellation occupies in the night sky. Nice coincidence that the first attack on Earth by the Borg came from the same direction as the major battle between Starfleet and the Borg.
The missile silo scenes were filmed at the Titan Missile Museum in Green Valley, Arizona. It is a decommissioned site and the only Titan missile complex open to the public. In order to get the shot of the launch they lowered a camera down the length of the actual missile on cables. Members of the crew arrived weeks early to get exact measurements of the warhead so they could build the cockpit "capsule" to fit over it. The money Paramount paid to use the site was used to build the new Welcome Center.
In the battle with the Borg Cube, there's a Millenium Falcon slipped into the film by Industrial Light and Magic. http://www.ex-astris-scientia.org/inconsistencies/curiosities2.htm.
Classic car nuts will find it humorous that a whole bunch of reproduction Chevrolet L-88 hood scoops are used as hatchway doors to the lifepods during the auto-destruct sequence, most obviously as they slam shut in close-up.
When Dr. Crusher is moving everyone out of sick bay when the Borg are about to break in, she calls for the holographic Doctor. She tell him to distract the Borg long enough for them to escape. He says "I'm a doctor, not a doorstop", which was a nice salute to a line often said by Dr. McCoy (which the Doctor also used in the Star Trek: Voyager TV show).
Not strictly a "movie mistake", but interesting to note. On the DVD commentary, Jonathan Frakes says repeatedly that the gun used by Picard in the holodeck scene is "a period German submachinegun". It is, of course, a Thompson M1928, possibly the most American submachinegun of them all.