Deleted Scene: A bunch, many of them containing a running gag about Buckaroo's life long arch enemy Hanoi Xan, the evil leader of the World Crime League who was responsible for the murders of Buckaroo's parents and first wife, Peggy. Penny even mistakes Lord John Whorfin for Hanoi Xan, which leaves him baffled and even asking out loud "Who's that?"
Executive Meddling - One reason given for the unexplained presence of a watermelon in Buckaroo's lab : Put in so the makers of the film could see whether or not the meddling executives had given up yelling at them as of that point in the production. Yes, seriously.
Fan Fic: In the mid-90's, author Ernest Cline wrote a script for the sequel tagged at the end of the film, Buckaroo Banzai Against The World Crime League, that was well-received in many Banzai fan circles.
BattleTech had a "Team Banzai" mercenary unit that was basically Buckaroo Banzai with the serial numbers scratched (not even fully filed off), including the watermelon.
Screwed by the Lawyers: The sequel was allegedly blocked by the rights holder for fear that the associated paperwork might uncover his creative bookkeeping.
The film's box office woes didn't help either.
What Could Have Been: A continuation of the Banzai character has been teased a few times, but has never came to pass:
A sequel tag, in the style of the James Bond films, for Buckaroo Banzai Against The World Crime League, was placed at the end of the film, but the sequel never materialized.