The script called for Aunt Entity (Tina Turner) to drive a vehicle. All of the vehicles were built using manual transmissions, which Turner couldn't drive, so a car equipped with an automatic transmission had to be constructed.
Originally, the film was supposed to be about a group of children living without parents in the wild. They were trying to decide what adult character would find them, when someone thought of Max. After that suggestion, it became a "Mad Max" film.
Max's eyes are different; the pupil in his left eye is permanently dilated. This is a nod to Mad Max 2: The Road Warrior: When his car is forced off the road by Wez and Max crashes, he suffers a severe injury to, among other body parts, his left eye. The disparity is easier to see in close-ups, and VERY easy to see in HD versions of the film. In the regular version, it's most prominent when Max first looks down on the Thunderdome.
The sandstorm at the end of the film was real, and a camera plane actually did fly into it for some shots. The storm in its entirety hit the crew in the desert, forcing them to ride it out in their cars and wherever they could find cover.
Jedidiah's airplane is a Transavia PL-12 'Airtruk', a single-engine agricultural biplane designed by Transavia in Australia. First flown in 1965, around 120 had been built by the time this movie was made.
George Miller lost interest in the project after his friend and producer Byron Kennedy was killed in a helicopter crash while location scouting. That may explain why Miller only handled the action scenes while George Ogilvie handled the rest. The film is dedicated to Byron Kennedy.
In interviews about Mad Max 2: The Road Warrior (a.k.a. "The Road Warrior"), George Miller said that while Max's world was after the collapse of the social/political/economic system we know, it was not post-World War III. However, "Beyond Thunderdome" explicitly contradicts this.
The trivia item below may give away important plot points.
Two scenes were cut from the film to bring down the running time. One where he dreams of his murdered wife and son, wakes up and cries. He realises he's become just as bad as the animals he used to hunt down as a cop. The other is Max takes a dying Gekko to the top of a sand dune at night, sees the lights of Bartertown and tells him they've reached Tomorrow-Morrow Land. A few seconds of this scene are included in the music video for Turner's "We Don't Need Another Hero".