To be able to make a NPC version of yourself that other players can battle like in RS & ORAS so that other players can battle would be a great feature I think, so that you don't have to be online a server whenever you want to battle your friends.
https://bulbapedia.bulbagarden.net/wiki ... ration_III
https://bulbapedia.bulbagarden.net/wiki ... ecret_Pals
Records of Win/Loss records and streaks could even be kept on the Gym Statue decoration item or something else appropriate. It would be very cool if you could keep records per team of Pokemon, to see if a specific combination of Pokemon are more difficult to beat than others, and what team the challengers used to beat your team.
The ornaments are all cute items that could be added just to add some fun to the game. I particularly like the breakable door, slide, stand, and tents.
https://bulbapedia.bulbagarden.net/wiki/Ornament
The posters could be added to the paintings in the game.
https://bulbapedia.bulbagarden.net/wiki ... on#Posters
https://serebii.net/omegarubyalphasapph ... ions.shtml
Capture the flag could be implemented from these. Secret base items could also be a great way to show off the Trainer Card you have to other people.
Additionally, I've seen a previous suggestion to add dolls of Pokemon to the games and the games already have plenty of those.
Pokemon Dolls from Gen 2:
https://bulbapedia.bulbagarden.net/wiki ... 9mon_dolls
Pokemon Dolls from Gen 3:
https://bulbapedia.bulbagarden.net/wiki ... on_dolls_2
Pokemon Dolls from Gen 4:
https://bulbapedia.bulbagarden.net/wiki ... on_dolls_3
Pokemon Dolls from Gen 5:
https://bulbapedia.bulbagarden.net/wiki ... _Catalogue
Pokemon Dolls from Gen 6:
https://serebii.net/omegarubyalphasapph ... ions.shtml