1. Explore every region ever created :]
2. Have achievements outside of the normal gameplay [such as explore regions, catching x amount of Pokemon, battling x amount of trainers, getting gym badges from each region]
3. Co-op traveling [party] - What if a duo players could travel together through a dungeon/tunnel [maybe a follow option so they can keep up with each other], and when a player enters a battle, they tag team against a horde of Pokemon. That could add a more interactive multiplayer experience rather than just a complete the storyline 'solo' game when traveling together. Maybe there could be a custom map with harder hordes and even a boss Pokemon or super-NPC-trainer at the end of the tunnel [and a prize for beating the 'dungeon'].
4. Gyms - What if the players could fight their way to the top and claim title of a Gym for an X amount of time [Not sure how this would work [because players are constantly logging in and out so it would require a lot of thought process behind this idea]
5. Trainer profiles [In the new X and Y] you can customize your trainer profile and answer questions about yourself.
6. In Soul-Silver version, you can turn and talk to your Pokemon [and they interact depending on where you are such as the beachside or the icy mountain] and it increases friendship /or decreases it can find shiny leaves and items on the floor etc. Perhaps adds +2 HP to the Pokemon if you max out happiness so that we are more inclined to have a happy pokemon + have more fun interactions.
These are just random thoughts off the top of my head more may be added later