After a contract dispute with the Disney company over likeness rights, Robin Williams agreed to return for this film as the voice of the Genie. As a result, many of the ideas and gags that were not used in Aladdin (1992) could be retried in the making of this film.
Among the Genie's many transformations is an old, grandmotherly woman, clearly meant to be "Mrs. Doubtfire" a character played by Robin Williams in the film of the same title, Mrs. Doubtfire (1993). The character originated as "'Andy Kaufman''s Grandmother" at his famous Carnegie Hall Show and was adapted for the movie.
The first idea for this storyline involved a villain from the Aladdin (1994) TV series named Mozenrath. The plot would reveal that the hero and villain were brothers. However (according to one of the writers) everyone thought they should steer away from the TV show. That and the fact that they couldn't get in touch with Jonathan Brandis (Mozenrath's voice), prompted them to move on to another relative of Aladdin's - his father.
In the first movie, Aladdin makes a wish to become a prince. He ultimately gets his wish in the third movie. Since Aladdin's father is the King of Thieves, that would make Aladdin the Prince of Thieves.