Okay, here's a whole big truck load of ideas I've had as well as some ideas I've seen floating around elsewhere, the creators of which I can't really remember.
* First of all, much can be done with a altered opening crawl. So many details can be changed and the whole style of the film can be completely altered. Let me give you some examples on how this can be done with an example of an altered opening crawl for The Phantom Menace:
It is an age of tension in the GALACTIC REPUBLIC.
The nefarious GALACTIC FEDERATION, an alliance of
power-hungry war lords, has mysteriously appeared
and spreads it's influence across the galaxy. Once shielded
by the brave JEDI KNIGHTS the old republic now crumbles from
within and without.
Now, the mercenary Federation has blockaded the peaceful
world of Alderaan seeking to place the helpless world under
their domination. With war looming ever closer, the Chancellor
of the republic has sent two Jedi Knights to negotiate with the
Federation in the hopes that peace can be maintained...
* Okay, so there's a few things to note about that opening. First of all, the federation is no longer involved with trade. They're simply the claw of Palpatine, an organisation of mysterious origin seeking to conquer the galaxy. The old republic is already crumbling, so it's final downfall is already being orchestrated. This gives the series a certain darkness that is generally absent from the prequels, or at least the first two. Naboo is retitled "Alderaan", giving the planet some sense of relevance next to the original series. You could argue that Naboo doesn't look like Alderaan, but you have to realise that this fan edit would constitute and alternate canon. The more high-tech cities of Alderaan undoubtedly exist, but away from the magnificent and historical palace of it's royalty featured here.
* Also, some creative voice editing can be used to make reference appear to Alderaan actually in the film. For example Panaka mentions later that a senator from Alderaan is being considered for the position of chancellor. Use this sound bite earlier in the film and Panaka could in fact say "Alderaan has no army!" as opposed to "We have no army!". I think this works even better, because we know this to be true from the original series as well.
* Next of all, alien languages. The Neimodians are not the least bit intimidating. Some solutions have been suggested for this, but one of the best and most interesting I've seen is to take dialogue from the likes of Passion of Christ and Apocalypto which both feature obscure and (I think, not 100% certain) dead languages. This is a great improvement and gives the editor a lot of freedom when rewriting the dialogue of the Neimodians.
* Next, droids. Rather than speak any recognisable language, they should perhaps speak a garbled techno-babble like the probe droid of Empire Strikes Back. It might make them slightly more intimidating. Another positive change, I think, would be to recolor the droids to gray and their tanks from brown to black. Star Wars has consistently used conflict between nature and technology as a theme, but the naturalistic browns and beige colors don't exactly help emphasise that. If changed they could not only look more intimidating but also help reflect that theme, as they would stick out as very robotic and alien whilst crunching around the forests of "Alderaan".
* Here's a little detail I think could help improve the prequels connection to the original series. It's a just a little one, but instead of the Trade Federation ship using that odd aqua-like and rather gratuatious bubble thing to communicate with Amidala, they could instead use a screen like that used by Vader in Empire Strikes Back. Y'know, that scene where he chokes the admiral using the force. A flash of static, the familiar activation noise and then the reveal of Amidala; it would add a little bit of a familiar ambiance to the scene, I think.
* Jar Jar completely rotoscoped out. Yep. GONE. Along with the Gungans in general. No more.
* I may seem to be hitting the same button again, but I also believe that Naboo starfighters should be changed to gray or a more mechanical color. As it is, they look rather like toys more than tangible star fighters as in the original series. A color alteration could help with that.
* The young Anakin Skywalker is actually a very quiet boy. A very quiet boy. In fact, he's so quiet and introverted and thoughtful most of his lines are no longer appropriate. He's tuning into the force, or something. Any reason at all. Anything to justify removing a bunch of his lines.
* Perhaps Wato should be given the Neimodian treatment too? Except his lips may be too articulate, unlike the Neimodians...Hmm. It would certainly work, as he could make more references to the treatment of slaves in order to increase the sense of villainy. Maybe.
* All lightsabers, excluding Sith lightsabers, should be changed blue. Why? Because then Luke's new lightsaber in Return of the Jedi has some meaning. It's the beginning of a new dawn for the Jedi. Whereas in the old republic they used blue sabers, in the new republic they will use green just as the founder Luke did. I think that would be neat.
That's all I can think of for The Phantom Menace. I was thinking that maybe Dooku could be introduced via Holograms to give his character greater purpose, but somehow I just don't feel there's space for that if the narrative of the first film is to remain uncluttered. Also, I quite like the idea of Grievous being the remains of Darth Maul. Perhaps he could even have Maul-like symbols pasted onto his body? Maybe? I just feel that Maul needed better treatment. I was expecting (and I think fans and the public were expecting it too) that Maul would be the primary antagonist of the prequel series, before he was cut-down in his youth rather like poor, poor Fett.
Anyway, there's my ideas....