I kind of feel like this topic was made for me, because my current--and largest--project is a rewrite of the second Digimon anime series. Though the season is my favorite out of all of them (not for quality reasons, but because I just absolutely love the six adorable human misfits and their monster buddies that are the main characters), I will readily admit that the plot is a mess and could have been given a little more care. That's what I hope to do with my rewrite of it: give the six adorable misfits* a wild ride that's coherent.
I've said it before, but I really want to read this. >_>
Digimon Adventure 02 is also the first one that comes to mind for me. It's certainly not as close to my heart as it is to Asty's, but when the character of the Digimon Emperor/Kaiser is at his best I almost wish it were.
Almost. Loads of plotholes, hole arcs that just fall flat, and the crest assignment is absurd IMO (crests are a Digimon thing that I won't get into here). It really needs a complete overhaul, but Asty's more the one to do it than me because you need some serious love for that kind of endeavor.
Aside from that, I'd rewrite Final Fantasy VII and VIII; VIII moreso. There are some terrific ideas at the cores of each--especially VII--but execution is lacking. Lot of things need to make more sense, and most critically in VIII Squall's development and his romance with Rinoa needs a lot of work. They rely too much on bonkers not-really-time-travel stuff to add gravitas to their love story when what they really needed was to give them more believable chemistry.
There is a remake of FFVII coming up of course (someday), but my main worry is that they'll leave the plot points intact while most of the changes go to the graphics, gameplay, and tone. If I were in charge I'd actually leave those latter 3 almost unchanged and instead rework the plot and polish up the translation.
Any rewrite I do for anything would be from scratch, btw. I think if you're going to rewrite someone else's work, it's best to explicitly make the entire thing your own take on their original story rather than do a subtle patch job. That way the original remains the original and yours remains yours and credit is less ambiguous.
EDIT: almost forgot; the reason I write Pokemon is basically the same as other posters have given. It's such a rich, expansive world that just begs for new stories to be added to it. Heck, you could write at least a short story on any of hundreds of Pokemon evolution lines and find something new to say every time. And by this point most of the official media is just professional fanfiction anyway, so it feels less like you're stepping on any specific creator's toes by taking aspects of the canon(s) in your own direction.