This I doubt will ever happen, but just for fun.... Who would you like to see take over based on their writing style? Brandon Sanderson? He's the first that comes to my mind.
This is also a good exercise because it might turn some of us on to authors to read that we'd never thought of.
Context matters a lot for me; in this hypothetical, is Jim handing off a completed outline? As in, "Here are all of the plot details I've been dropping cryptic hints about for twenty years," or is this a case of (knocking on wood, crossing my fingers, throwing salt over my shoulder) an unexpected...demise...on the part of Mr. Butcher?
If the former, fine. If Jim just got fed up and handed the series off to someone else, but part of the agreement was that they follow Jim's outline, I might--MIGHT--accept someone else, so long as the plot was in competent hands. That's really what I care most about; plot competence. Kevin Hearne, of the Iron Druid Chronicles, has plots that feel like they sort of just
happen, though I do think he understands the importance of setting and mechanics; he's very fond of Magic A is Magic A, which I appreciate. Dean Koontz puts together plots with pretty solid detail, but can't really get by on just plain English, which is what I prefer, and most of the supernatural elements of his stories (like Odd Thomas) seem to run on whimsy and contrivance. Neil Gaiman, after reading American Gods, is, in my opinion, an extraordinarily competent writer, but left me with such a disappointingly anticlimactic ending that I wouldn't trust him with the Dresden Files as far as I could punt him underwater.
It'd have to be someone who is a genuine fan, someone who is insanely devoted to the story, setting, characters, and mechanics, and someone who can mimic writing styles, or at least write in plain English. (Personally, I don't like writing styles like House of Leaves or Fight Club, because they spend so much narrative time describing things in ludicrous detail; it feels like I'm running on a treadmill rather than moving forward in a story. I prefer narration to poetry, if that makes sense—thus, plain English.
Get to the point, I say!)
If it's the second case, where Jim suddenly passes away, then the only ones who can touch the series from then on should be his kids, and that should be in a single volume that ties up loose ends based on Jim's notes. At that point, I wouldn't care about writing styles, I'd just want things to be ended.