Beam Me Up, Scotty!: A minor one, but Anakin is sometimes associated with the line "This is so wizard!" or variations of the line; while the line is in the film, it's actually said to Anakin by Kitster.
The Cameo: Warwick Davis (who played the main Ewok in Return Of The Jedi) had a couple of minor (but visible) background roles.
Cast the Expert: Darth Maul was played by Ray Park, who was a martial arts expert and stunt man, not an actor, although he has had a number of acting roles since then. This is one of the reasons that he had virtually no lines, and what lines he had were dubbed by Peter Serafinowicz.
Creator Backlash bordering on Old Shame: Jake Lloyd isn't fond of his role in the movie, thanks to having been mocked about it since childhood. He went through a phase where he destroyed all of his Star Wars memorabilia and refused to talk about it. However, an interview with a Star Wars fan site has confirmed that he's mellowed out since then and contrary to popular belief, he does not hate Star Wars after all the bullying.
Deleted Scene: This movie, like the others, has its fair share of deleted scenes. One of the more notable ones is where Qui-Gon and Anakin start to depart from Tatooine. One of Maul's sith droids attempts to tail them, only for Qui-Gon to suddenly slice it down, and then explaining they need to hurry back to the ship. This scene explained why immediately in the next scene, they were running back to the ship when the previous scene had them walking calmly from the market.
Another notable deleted scene, taking place after the podrace, involves Anakin beating the crap out of one of the other kids because he accused Anakin of cheating. Would have foreshadowed Anakin's prone to anger. Anakin's victim? A young Greedo.
Easter Egg: Aliens from E.T.'s species make a brief, split-second appearance in the Galactic Summit scene.
Fan Nickname: Fans critical of Jake Lloyd's performance as Anakin sometimes call him "Mannequin Skywalker".
The Japanese-dubbed version is an interesting case: Unlike the sequels, all the movies had the same voice actors, regardless the media and networks (and avoiding The Other Darrin trope that the original trilogy suffered in Japan for years). In fact, the only voice actors switched between movies were Palpatine and Jar Jar Binks's VAs, who died between Attack of the Clones, Revenge of the Sith and the animated series.
Recycled Script: The heroes visiting an underwater city to parlay with its ruler was originally a big plot point in Lucas' first draft for the original film, The Adventures of Starkiller: The Star Wars.
Sequel Gap: 16 years between this and Jedi's release. The lead actress was two the year the last one came out. Also, if Episode VII comes out in 2015 as planned, it will have been the same number of years since Phantom premiéred.
Spoiled by the Merchandise: The soundtrack to Star Wars Episode I The Phantom Menace was released before the movie itself came out. Anybody who picked it up got treated to tracks called "Qui-Gon's Noble End" and "The High Council Meeting and Qui-Gon's Funeral." Oops.
Star-Making Role: Both Ewan McGregor and Natalie Portman were well respected actors but were in either small roles or independent films. This movie made them big names. Ray Park also fits in, as his presence in the final battle was one of the most memorable parts of the movie.
On the other hand, it was a Star-Derailing Role for Jake Lloyd. He was never a star, but he had a respectable career as a child actor before this film. He did one movie after it and then quit acting. He said in a recent interview though that he's pursuing a career in film editing, so he isn't totally gone from the movie-making scene. One can only hope he has/had access to Wil Wheaton as a mentor.