Something not mentioned, but being able to see what other pokémons are in the gym you are holding. Or what level it is. Something like that.
A friend of mine, choose purely at random his team. Now he regrets it because most of his/my friends are in an other team. Maybe just like the namechanging option, a teamchanging option?
Cool! Maybe besides team rocket, some NPC's that do something with your Team. I love the teamvibe offgame, but don't sense it ingame.
Would be super handy! Nice idea.