I'd like to see HHH kick-off an episode of Raw -- whichever episode they decide to lock in a full-time GM -- by introducing Jim Ross as the new general manager.
From there, I'd like to see JR say that the days of meddling, intrusive general managers are over. He could announce a few top-tier matches for that night (something that crowns a contender for the Intercontinental title, maybe the tag titles, and maybe a World Championship match or something to make it feel special), then say the days of 20-minute announcements are over with, that anything he has to say can be done from his office. Then, the only time we see or hear from him is backstage cut scenes where he's either discussing something with a talent or announcing a match. Every so often he can walk out onto the stage when chaos has broken loose in the ring to book something, but otherwise he remains mostly behind the scenes and out of the main storylines.
The thing about most of the possibilities -- Regal, Stephanie, JBL, Foley -- is that they all have distinct personalities and histories. JR is more or less neutral. He can play the role of the arbitrator without constantly butting in with his own agenda. He can open shows by announcing that Dolph Ziggler will face John Cena and Daniel Bryan will face The Miz, not because he has some vested interest but because he thinks those are matches the audience would like to see. There would be no reason to have 3-on-1 matches and endless contract disputes and stipulations about people getting fired because those wouldn't be where his interests lie. His interests are instead on putting on the best show possible. No cheap heat, no alliances, no agenda. You can still have drama and storylines and lengthy promos, but they can be cut by the actual talent roster rather than by some new incarnation of the Mr. McMahon character.