Thanks for asking, I had a lot of fun typing this out. I'm one of those people who has pretty strong opinions about what these movies could have been, but I'm also not a professional in the film industry, soooo...
(By the way, you should answer your own question!)
Thinking aloud as I go here.
The structure of the prequel trilogy as a whole was out of whack. In the OT, ANH is a standalone adventure that contains complete character arcs for our heroes, manages to introduce a universe, and wraps up all the immediate conflicts. It leaves the door open for a trilogy, but everything in the room is sorted and tucked away. TESB is a great second act. It's darker, and as the middle entry, serves to lead directly into the third act.
In TPM, there was too much politics and it didn't feel like a movie that could hold up independent of its successors. While the Naboo plot was tied up, the characters were bland, I had no idea who the villains were or what they wanted, and the end of the movie felt like a giant ellipsis.
I'd get rid of the phantom menace completely. No need to introduce Sidious if it's too early, but have an identifiable villian. I love the notion of having Darth Maul survive into the later movies, giving Obi-Wan deeper motivation to pursue the villains. (This was Belated Media's idea, not mine.)
To answer the question: I'd have TPM start with a simple Jedi mission involving Obi-Wan and Anakin (already adults). Its purpose is to establish their rapport and the boundaries of their teacher/student relationship through action. Maybe the mission's a failure and that sets the rest of the movie in motion, or maybe it's a success. Or, maybe it has nothing to do with the main plot. The purpose is to set up their back-and-forth without the cringe-inducing remarks about "that time you fell into a nest of gundarks."
I'd love the bulk of Episode 1 to be a rescue mission, despite the similarities to Episode IV. Throwing Anakin and Padme into the crucible early is a great way to foster chemistry and attraction. They don't have to consummate the relationship just yet: the purpose is to establish chemistry between the three protagonists. If it's not a rescue mission, then IDK. Padme needed to be in on the action much earlier, because in Star Wars, it's the action that moves the plot (and characters) forward. Having Anakin be her white knight early on would set up their romance perfectly as a contrast to the later films, when he is unable to save her despite his growing power.
Also, I'd have Anakin start to fall to the dark side in Episode 2. I know that's supposed to be the big moment at the end of the PT, but Anakin's decline needed to be accelerated to make Episode 2 darker. Put Anakin through the wringer to bring out his rough edges, end with Padme's fate in question, and have that ultimately resolved in Episode 3 with Anakin being unable to save her. Love is the biggest motivator, but I didn't buy Lucas' version of it at all. If Episode 1 establishes good chemistry, and Episode 2 makes Anakin/Padme Facebook official, Episode 3 is where it comes apart.
Final idea: In Episode 3 Obi-Wan is forced to choose between saving Padme and achieving something else, a la "The Choice" from Toby Maguire's first Spiderman movie. I'd love it if it went down like this: Obi-Wan is put in a position where he can save Padme or not, Padme makes the decision for him, but we see that Obi-Wan was so attached to Padme that he wanted to save her. Anakin shows up at this moment, assumes Obi-Wan let his wife die, and in a fit of rage and grief, attacks him. Big duel, Anakin becomes a Sith because now, instead of the immortality mumbo-jumbo, he actively hates the Jedi. They were unable to save the love of his life.
Giving Obi-Wan some complicity for Padme's death gives extra weight to his moment of self-sacrifice in Episode IV. Obi-Wan is now a man racked with grief over the death of one of his best friends.
All the circumstances I've described make a lot more sense to me than the PT, where Anakin and Padme make googly-eyes for two films, then he deliberately attacks his wife and blames Obi-Wan. It was all hand-waving by Lucas to rush us to the final big encounter. It didn't have to go down my way, but the bottom line is, Lucas didn't put the characters through the wringer and have their arcs develop as a result of that.