I think there are two things contributing to the staleness of the current game.
First, just because changes in a computer game can be made quickly doesn't mean that they should be. I think a model sort of like the one ccg's (collectable card games) use would be better. The cards in a set, once printed, are hard to change. Bans and such are sometimes handed out, but only in the most drastic situations. Luckly, in lol, teams are given 3 bans to issue each match so the urgency to nerf is lessened. Changes should be made less frequently, and on a set schedule - perhaps once at the start of a season, and one in the middle to shake things up and correct balance issues. Currently, things change so fast that the players don't have time to adapt. When a champ becomes popular because of how it works against the current meta, Riot runs in quickly saying, "Don't worry guys, you don't need to come up with a strategy to counter this, or even use one of the bans you're given. We got this." Like in ccg's, the idea should be, "This is the current environment, adapt. If there is an immediate need, we'll deal with it, but other than that, use the tools we've given you to deal with it."
Which leads into the other issue I see - the tools. We're limited to six slots, so offer lots of item choices that fit different situations. Right now builds are pretty cookie cutter, but what if there were more choices? AD plus armor, AD with MR, AD with AS and health, and all kinds of other combinations? Let there be be a meta game within the game around which team geared up the best to go against the current opponents team. I think this idea would also benefit from more items with activated abilities, but we could start slow and just add more build options.
Conclusion: the state of the game should be more stable to allow players the time to come up with counters, but those players should have access to the tools necessary to make those counters possible.