Especially with the benefit of hindsight. I feel like some of the prequels actually worked, but it was overshadowed by the obvious nonsense.

CAST/CHARACTERS

I'd keep:

• Ewan McGregor as young Obi-Wan
• Samuel L. Jackson as Mace Windu
• Natalie Portman as Padme
• Ian McDiarmid as Palpatine/Sidious

I'd re-cast:

• Anakin. It's a tough balance to hit: you have to be both believable as a young/naive/high-potential young Jedi and believable as "this dude eventually becomes Darth Vader.* After some considerable thinking (and potential circlejerk notwithstanding) I'd have to go with young Tom Hardy, and I'm convinced he would have nailed the exact vibe perfectly. Hayden Christensen put it in the eyes; Tom Hardy puts it behind them. And he'd be less "tortured Jedi," more, "if Padme dies I'm going full Darth Vader."
• Yoda. Bring back the puppet. I don't care how goofy the mouth looks, this is the I'm distracted by the obvious 900 year old wisdom in the eyes because they nailed the face in 1980 and don't have to change a friggin thing.

I'd remove:

• These noseless baffoons
• Jar-Jar Binks, obviously.
• Chewbacca's appearance in ROTS. We don't need to tie up every loose end. A whole planet full of Wookiees and only one--the sidekick of a smuggler--has ever done anything?
• Qui-Gon (sorry, Liam Neeson).
• Count Dooku (Christopher Lee is admittedly a badass but you'll see why)

Episode I: The Phantom Menace

In case you forgot, the Phantom Menace is basically: Jedis help Queen Amidala with her political situation, they land for repairs and get into hijinks and discover the future Darth Vader, political stuff on Coruscant, climactic battle. So much more could be fit into this! It's an episode of the greatest saga of all time!

Keep:

• Some people cut this movie entirely out of their re-watch, but I think the Anakin origin story is important, not to mention runs parallel with Luke's journey in ANH, making Luke's ultimate redemption in ROTJ more meaningful as a contrast to ROTS. It just has to be totally redone.
• Pod-racing, but liven it up with a completely different context.
• That's about it.

Remove:

• The Trade Federation stuff. The plot of "A New Hope" was pretty cut and dry: rebellion wants to stop evil Empire from using its ultimate super weapon. The plot of Phantom Menace feels like an article from the Wall Street Journal. I'm still not sure I get it.
• Death of Darth Maul. That badass had to be badly damaged in the climax, but he's got to live and return in II and III as a villain we've seen before with a history and a motive. Imagine Darth Maul's storyline, except Darth Maul is motivated.*
• Qui-Gon. Obi-Wan should be a newly-minted Jedi on his own who's still naive and just figuring this stuff out, and his discovery of Anakin (and belief in his potential) will make his regret and guilt years later all the more poignant. George Lucas gives that burden to Qui-Gon and kills him in one movie. We only have so much time with these characters; Qui-Gon's screentime should be for fleshing out Obi-Wan.
• Any "kid" stuff with Anakin. Maybe he's even full Tom Hardy in this movie. Luke Skywalker started his training late in life and became one of the greatest Jedis of all time; there's precedent here, and the audience is given more time (three movies instead of two) to develop a relationship with the character as we know him in Episodes II and III.

Add:

• The Clone Wars start at the end of this episode, not episode II. The way Lucas does it, the Clone Wars are basically inserted as "War!" in the Episode III intro. Why skip that awesome material? Make the Clone Wars the backdrop for Episode II, in fact, call that episode the Clone Wars. Skip all of the political machinations in Episode I and use the "Padme-assassination" stuff, which is more interesting, from Episode II.
• A more obvious Palpatine/Sidious twist toward the end. It would be great if Ian McDiarmid could be somewhat in the background the whole time, just a genuine nice guy, and then we watch him transform into Darth Sidious at the end and go "ohhhh, so that's who that is!" Merely suggesting it in the real Phantom Menace was okay, but we need to get rolling here.
• The pod racing to the second/half two-thirds point, the escalating action that brings us to the third act, not as a gimmick to get Anakin off of Tatooine.

Revised Plot:

Before she can make an important appearance in Coruscant, Queen Padme Amidala is captured by a mysterious figure, who demands ransom. In response, Coruscant sends an emissary to make the arrangement; but also Obi-Wan Kenobi, should negotiations fail. They do. In his daring rescue of Amidala, Kenobi makes a troubling discovery: the figure behind the kidnapping is in fact DARTH MAUL, the lord of a faction once believed to have been extinct. The early duel scene is important to establish Maul's superiority to Obi-Wan; Maul is only subdued with the intervention of the rest of Kenobi's team, and even that was...a little too easy. Maul is kept on a tight leash and refuses to speak on the journey back to Coruscant. When the ship's hyperdrive suspiciously fails, destiny points them to Tatooine.

There, Obi-Wan negotiates for a new hyperdrive and is hustled out of fifty credits by young Anakin Skywalker. Obi-Wan follows him home, impressed that he didn't sense the con through the force, and finds that the cocky kid was only trying to feed his sick mother. When he tries to introduce himself, the shanty is invaded by men representing the HUTTS, and both the Skywalkers and Obi-Wan's team (including Padme, who was not left on the ship with Maul aboard), are captured and dragged to a snotty chamber. To Kenobi's astonishment, it is Anakin who saves the team through the spontaneous use of the force; unheard of anyone untrained in the art. Kenobi convinces Anakin to come to Coruscant to get proper medicine for his mother who comes along (foreshadowing how far Skywalker will go to save his family).

On the way over, Anakin is told to stay away from Maul's cell; he can't resist a peek later, though, and when Maul finally looks over, he closes the door.

Back on Coruscant, Padme does her political thangs and the Jedi Council is mixed on Anakin, much as they are in the real Phantom Menace. Eventually it's decided that he's too much of a superstar to leave without proper training. Anakin agrees because he wants the power to help his mother (yada yada foreshadowing). Anakin and Padme, both young up-and-comers in the big city, bond over their commonalities.

Maul, plotting his escape through a few quick-cut scenes earlier, finally sees his plans come to fruition and receives orders from a mysterious DARTH SIDIOUS . He destroys everyone in his path which includes, unfortunately, Anakin's mother, and after pursuit by the Jedi, escapes to the crowds surrounding a POD RACE.

The Jedi are hot on his tracks when Anakin, out for revenge, intervenes and inadvertently gives Maul an advantage: Maul gets access to his own pod. MAULPOD. After a final battle on a shuttle launchpad in which Maul kills several anonymous Jedi, Maul escapes. Mysteriously, the Coruscant defenses let him go through. Wonder who could be responsible for that.

I'll do more if there's any interest, this is getting long.