What I want? I want to, when surfing, be able to look around me and see things that are not within the tiny little field of vision, like a giant stone tower looming over everything (cough Emerald cough) or a huge sand bar with trainers and items on it, instead of having to blindly bump back and forth between walls of stones until I stumble upon something of note. Being able to use a proper camera on other routes would also be nice, but usually on land the camera gives you the views you need.
Starting with your pokemon would also be nice. It's a small thing, but it always feels so impersonal to get your pokemon at the start of the adventure from a professor, having never seen them before. What if you're introducing a new kid to town, and there are the three starters next to you and he asks you which one's yours? And then asks you if it has a name?
What if the TMs you get from gym leaders are replaced with HMs? Cut, for example, might be the Grass type gym move, while strength might be from the fighting type gym, or the normal type gym. A ground gym would have something like Rock Climb, and so on. HMs should be unnecessary for completion, but instead allow access to new items and places, or provide more mobility in traversing routes. This keeps the gym leader's connections to the moves while making it make sense. Also, the HM moves could stand to be more varied, or change type, from the normal fare. Cut being Grass type, for example, or a ghost type move that lets you pass through certain obstacles.
Oh, also, gym leaders might be defeatable in any order (or at least there would be some freedom to it) and have their teams scale somewhat as well. Maybe every ten levels you go up, their team improves.
Your rivals' levels should scale with yours. It would alleviate the problem in the rivals often being either way too hard or way too easy. Ideally, their starter pokemon's level is equal to the highest level in your party. Maybe the rest would scale to match yours as well, or maybe they'd trail at fixed points behind their starter. If you only have a couple of pokemon, maybe their levels scale down a bit, or maybe they only use as many as you have. It would make them be more of an appropriate challenge, as you have to use superior tactics. Instead of powerleveling, you have to decide at what point your team is going to be better than theirs- do you evolve your starter to get the stat boost and moves and risk them doing the same, or do you leave your team lower leveled and rely on TMs and healing items to stay ahead?
A difficulty setting, if implemented, might affect how the level scaling works- an easy mode would lower the rival's levels from you by a small bit, while a hard mode might set them as many as five ahead per pokemon. Gym leaders would also be affected by this, as in BW2.
One last idea: what if E4 members focused on specific techniques or tactics or battling styles instead of a type? For example: a specialist in priority who uses moves like Bullet Punch or Avalanche, or someone whose whole team knows baton pass and a boosting move? Or a team built entirely around playing with the weather, with one mon for rain and one that benefits from it, another pair for sun, and so on.