Gen II and its remakes - I agree with more involved day/night cycle and weekly events, and Apricorn balls as well! Plus, the Pokegear and the events to get major characters' phone numbers (such as those of Gym Leaders). Following Mons like in HG/SS too, with reactions to their environment!
Gen III - Again, the Match Call function. Being able to rematch the Gym Leaders during the story like in Emerald and having them get stronger alongside you would be nice. It made the Hoenn Leaders feel more realistic than in OR/AS. The Fame Checker from FR/LG was great as well, but the League Cards from Sw/Sh more or less filled the same purpose, so...
Gen IV - POKÉRADAR. I got so. Many. Shinies! from it in Platinum! And bring it back just as it was in that game or in BD/SP, not like in X/Y. A Battle Frontier more reminiscent of Sinnoh's than of Hoenn's would be awesome as well.
Gen V - Agree on seasons. Also, different themes for each Gym.
Gen VI - Better customization options for both genders. No games have managed to match the options we had in that game. Same goes for the post-game episode with Looker; I have yet to see a postgame story more satisfying than that one.
Gen VII - SOS Battles and open-field Trials instead of Gyms. Don't get me wrong, I love Gym puzzles, but it feels less claustrophobic to complete them in an open space that takes advantage of the existing world.