Well, given Harry's status or lack thereof with the White Counsel, it's got to be someone already estranged from the White Counsel and/or has no interest or hope of getting in their good graces.
...
I suspect Harry has no useful way to get a candidate to the WC's attention. Anything Harry does -- anyone Harry recommends -- arrives under the taint of "probably the tool of that fae-compromised warlock, Dresden." Harry would be putting them into the same "Sword of Damocles" Warden-hounded role that he hated so badly (if they didn't just chop the poor 'prentice out of hand, on "general principles").
So anyone he finds who really needs training?
100% on Harry for that training.
... I also think that Butcher is unlikely to intro a major new character at this point ...
Tho FWIW it doesn't have to be a "major" new character! IIRC we only ever met Kim Delaney in the opening of Fool Moon, but she had been (for a short while) Harry's apprentice; just enough to set her on her feet, she was never going to be WC-caliber. So it might be another short-term apprentice like that.
Also, I think it highly-unlikely Jim will intro & lay the foundations for a major/recurring character in one of the shorts: he usually keeps key character & plot development to the novels.
Looking back, I'm going with Fitz from Ghost Story. Certainly seemed like that character was going somewhere. Took up a subplot and a lot of ink but then exited midway thru that book, and haven't seen him since. Fitz felt like groundwork, and you know how Butcher drops stuff in books from left field and then uses the stuff later.
All told, I really like this theory!
Your points about "subplot... a lot of ink... felt like groundwork" are I think pretty damned compelling arguments that Fitz
will be back; and this could be his (re-)entrypoint.
... I think Mort Lindquist would be more suited to act as a mentor to Fitz, seeing as Mort can interact with ghosts he could give Fitz much more practical advice then Harry ...
How about this plays instead to a different fantheory: Harry organizing the "lesser talents" (via the Paranet) not just for self-protection, but to police their own ranks (vs. warlock-ism), to investigate, etc; but above all to detect emergent talents and scoop in to educate them, train them, and as much as possible to prevent the Wardens from needing to Snicker-Snack any more youth-gone-astray.
Mort would be the natural "department head" for all sorts of ghostly stuff; so (for "Out Law") Fitz is Harry's apprentice, but then (near the end of the story) Harry hands him off to Morty for specialized training, and Fitz remains through the following novel(s) an ongoing (but peripheral/occasional) character.
Harry finds that this works
really well; he does a compare/contrast between "Korean Kid" vs "Fitz" and decides to try formalizing & spreading the notion.