I prefer to think Alien3 never happened and that it was just some supremely messed-up dream Ripley had while in hypersleep, but if they really had to make a third installment.... yeah, I would've done it a lot differently. It's not a bad film per se; after all, it was a worthy directorial debut for David Fincher, but it was bleak to the point of being ridiculous (blame it on early '90s angst?). It was disconcerting they killed off Hicks and, more pointedly, Newt, thereby throwing away everything Ripley had fought so hard for in Aliens. And the prison planet made for an interesting setting but lacked the sheer claustrophobia of the first chapter or the relentless they're-coming-out-of-the-walls terror of the second. I would've preferred to have Ripley return to Earth and finally do battle with the evil Company, and of course at least a few of the acid-bleeding beasts... who's to say there weren't other Alien Queens elsewhere in the galaxy and the Company happened to have captured one? I'm not sure where Hicks and Newt could fit in the story but if nothing else, the Company could've kept them hostage in hypersleep.
I've always said that in the third movie the suits forgot who the real star of them was, and wasn't Sigourney Weaver, as good as she was.
If they absolutely had to do the prison planet thing, I would have a grown-up Newt working on some sort of space freighter or other type of spaceship. There could be a nice scene where she talks to her 'parents', Ripley and Hicks, on the vidphone, perhaps Bishop as well. Then the ship crashes on the prison planet and all of her crewmates are killed. Rev up the plot of Newt realizing that the xenomorphs have arrived on the planet and trying to figure out how to do battle with them, calling on her half-forgotten childhood memories. There would be no "OH NOES!!! I'm preggers with an alien baby so I'll throw myself into a volcano!" stupidity at the end, just Newt using her wits and ingenuity to battle and destroy the xenomorphs and escape from the planet by the skin of her teeth. Maybe we could have a homage to the original movie and she could rescue a dog along the way.