If the games could support it, I would love to have a huge list of moves you could use outside of battle at any time, possibly for different effects. Cut a tree down with Cut or burn it out of the way with Flamethrower, things like that.
On a related note, I would love to have the Moves system overhauled so that Pokemon would be able to use certain moves inherently. At the very least, replace Struggle with Tackle; I know it's a game, but it messes with my sense of logic that my Water Pokemon has to be taught Surf, or a bird Fly, etc.
As for actual HMs or other current out-of-battle moves, I'm hoping that Gamefreak eventually decides to replace it with some kind of tool system. The Player (not the Pokemon) uses a Lantern in place of Flash (and possibly Defog, if that ever comes back?), an Axe instead of Cut, a...Hammer instead of Rock Smash. Some moves can't be replaced with a tool that easily (Jetpack for Fly? lol), but having to have an HM slave -which takes a slot from your party- or weak moves -which takes a slot from your moveset or can mess with your battleplan- or traveling back and forth to a PC is just...a nuisance.