Movedex: Suggested by Victreebong of Serebiiforums, this idea would focus on using all the moves in the game and registering them to a movedex.
This idea has potential, as not only there are several moves exclusive to event Pokémon, there are also several moves that are event exclusive period, such as Happy Hour, both of which will increase as the franchise grows.
Pokémon Racer: A perennial idea that became popular with the reveal of Rhyhorn racing in XY, this idea have existed since Gen I, when in OS animé, we saw various species racing.
This idea can work in various ways, either as in horse racing, or in dog racing.
Nurses: While I have little idea how such as sidequest will be outside of player taking missions to collect and heal randomly generated Pokémon, there has been an increase about the role of the nurses in the games.
We have several healer Pokémon, such as Audino, Comfey, and Happiny-line.
The player character's mom in B2W2 was a nurse.
And nurses have been a presence in the anime, not just by Nurse Joy, but also in Jessie's backstory.
Item Collector: Serving as an ItemDex for the game, this sidequest would involve completing all possible items in the game.
This idea actually has some potential, as there is an abundance of event only items.
Ruin Maniac: Just a mission sidequest, this idea would focus on player character entering old relics, either to find old items or encounter old Pokémon.
This idea was already with Johto's Unown Dex Arceus Plates and Unova's Relic items, the difference would be that this idea would provide encounter areas ancient artificial Pokémon such as Baltoy and Golett who are generally rare and forgotten.
Botanist: Botany related classes have always existed in Pokémon, and berries are a major feature since Gen II. But there hasn't been any major idea of collecting flowers or other plants.
This idea would also help out to buff up the amount of Grass-types.
Inventor: Another idea that has been since Gen I, so far outside of Shell Bell, berry related items or Apricorn Balls, there hasn't been any major Monster Hunter or even regular JRPG-esque item crafting.
This idea would also help to expand the Electric and Steel-types.