My choice would probably be Die Hard. Such a great setup, fantastic hero, and one the greatest villains of all time very nearly ruined by the relentless, cartoonish idiocy of the police. I'd have McClane unable to raise the alarm for longer, then have the cops act sensibly but be stymied by Gruber's brilliant planning and ability to improvise.
Oh, god, yes. The cops in that movie are entirely cretinous, with the apparent exception of one donut connoisseur.
If I couldn't do it with the young Bruce Willis and Alan Rickman, then I'd choose Citizen Kane and cast Chris Hemsworth and Scarlett Johannson and put in a bunch of explosions, sex, and possibly sexplosions.
You didn't say the goal was improving on the original artistically.
I don't understand; in what way are sexplosions not artistic?
A movie I'd probably try to fix, if I could: Stranger than fiction. The setup, of a person who discovers he's a fictional character living in the real world whose author plans to kill him in the end, and who sets out to find his author to prevent his own death, is pretty well handled. However, the last act (spoilers), in which Our Hero decides to let himself be killed in order to preserve the artistic integrity of the story that he's in, while every actual person around him lets this happen because apparently critical response to a novel is more important than someone's life, is a truly horrendous failure to pay off what they set up, complete with cop-out not-really-killing-him-after-all "twist". That Emma Thompson's character apparently possesses a magical typewriter, upon which whatever she types happens for real, is entirely unexplained in the story, and utterly unremarkable to any character who finds it out.
And, if I couldn't think of an actual ending for that movie that wasn't infuriating, I'd make the following small tweaks to Star Trek: Nemesis: cut the opening scene on Romulus; replace the transparent cymbals in the wedding scene with real ones; and trim the remaining scenes from after the wedding to the end credits. Sure, it's a bit short for a big-budget movie, but that's the price you pay to get quality story-telling.