The infamous "you killed your rival's Raticate" theory from the first games. Never played seriously nowadays outside of creepypasta.
It does still pop up, at least in various Nuzlocke runs, especially the more story-oriented ones. While it's still not overly common, perhaps with the players' pokémon dying being a core concept of the genre, the raticate might provide a nice enough callback to allow the theory one last grasp at life.
The names of Silver, Kris, Leaf, Barry, and Hugh may fall under this. The first three are named after their games, Barry is named after his anime counterpart, and Hugh is the character's default name.
Trainers all beginning their journeys at the age of ten. While this is canon to the anime, the games have made no such statement (and you can in fact battle trainers who are much younger, including preschoolers- though they usually aren't found in places that would indicate that they're out challenging Gym Leaders).
A "subtrope" of this is the main protagonists of each game being ten. The only protagonists to have a canon age are Red and Leaf, who are eleven. Hilbert and Hilda are implied to be much older.
The three Pokemon that died in the Burned Tower and were revived as Entei, Raikou, and Suicune are believed to have been a Flareon, Jolteon and Vaporeon respectively. This is backed up by the three having the Eeveeloutions normal abilites as their Dream World Abilities.
Less commonly, they're believed to have been three Eevee, and their revival is thought to have acted as powered-up Water/Thunder/Fire Stones.
There's one popular fanon that combines both the game's canon with the anime's canon. Giovanni having to do with Mewtwo's creation (anime) and that all of the criminal activities that Team Rocket does were to try and stop Mewtwo once he broke free (game).
Gym Leaders having multiple teams of Pokémon, with which they can provide an appropriate level of challenge to trainers of different levels. Canonized in Black 2/White 2 by Cheren's mention that he's not allowed to use his old team against you in your Gym battle against him, and is also mentioned in Pokémon Origins.