God Never Said That: For a long time it was shown on the movie's poster that Gigan could fire an Eye Beam from his forehead, above his eye, yet this ability was never shown in the film, though later appeared in Final Wars and the Atari/Pipeworks video games of the 2000s. Many fans speculate that this was due to budget reasons, evident by the abundant amount of Stock Footage in the film. It was later reveal however by Special Effects Director, Teruyoshi Nakano that the REAL reason it was never shown was because he didn't believe it fit with the character, so neither film demonstrated this ability for said reason. This all can be read here.
Additionally, Gigan also possesses the ability to breathfire, but was never shown to in the film for the same reason as above.
Troubled Production: This movie had a long and complex path from its original script to theater screens, with major revisions being made constantly in order to stay within the meager budget. See What Could Have Been for details.
The first idea had Godzilla and Anguirus "team up" with a Rock Monster named "Majin Tuol" to fight Ghidorah, Gigan, and Megalon. Other differences that were to take place was that Godzilla Tower was just a base of operations for the heroes, the villainous leader behind the monster attacks was a floating brain named "Alien Miko", and Majin Tuol's only existence was to kill the Space Monsters (with the exception of Miko, who is killed by Godzilla). Megalon would go on to appear in the next movie.
The second idea had Godzilla team up with Rodan and Varan to fight Ghidorah, Gigan, and a Dragon Monster named "Mogu". This idea was much more closer to the film that it is today, introducing the villainous Space Hunter Nebula-M aliens, as well as the monsters being summoned by tapes being played. The film also was to focus heavily on King Ghidorah, hence it's original title as "The Return of King Ghidorah". The idea was eventually dropped with the budget of making new suits for Rodan, Varan, and Mogu was too expensive (Varan's existing suit was already unsuitable for use, which necessitated a re-write of Destroy All Monsters), and Rodan's suit had been used in three movies in the 1960s) and instead were substituted for Anguirus, whose suit (Soshingeki Angira, made for Destroy All Monsters in 1968) was in good condition. This idea also was to give Gigan a Mace for one of his hands.