Auckland [8]
Bologna [18] (19-1) One of my favorite city states, and I think it should finish in the top 3, but it's not right for it to have nearly twice as many points as the second one in this game. Extra GPP is always welcome, but if you're not suzerain, it's not going to cost you a scientific or cultural game.
Cahokia [5]
Yerevan [11] (10+1) My vote for the best city state in the game. It's really on a whole other level than other city-states. Everything the other city-states provide is nice. It's nice to have extra GPP, or a bunch of free amenities, or extra production with a coastal civ, but those city-states themselves don't have the level of impact that Yerevan can provide. Yerevan literally wins games. As some others have pointed out, it doesn't merely just give extra things that you're making anyways - it gives you abilities that you won't have anywhere else in the game. You can have several Debator apostles on the map, wiping out enemy apostles like a super-charged wolfpack. Every apostle you make can have triple the spread or just annihilate enemy religion influence. And you can have multiples of these types. Even if you just rely on RNG, once all your good promotions are used, all your new apostles are stuck with pointless, weak ones like 100 gold on conversion, or converting barbarian units. It gives you the best bang-for-the-buck with your apostles.
No, it doesn't have much use for say a scientific game, maybe not much for diplomatic, but as others have pointed out, it has great uses for relic-based cultural wins and domination-via-Grandmaster's-Chapel. But, I would argue that just because it's not important in every game, that it does not mean that it doesn't hold amazing value. It does what it does and makes certain victories a walk in the park.
Here's a thought experiment: Assume we had a brand-new city-state in the game. And once you get suzerain of it, a little button pops up that says "Win Cultural Victory". And when you click it, in one turn, you suddenly have more tourism than anyone else and win a cultural victory instantly. Pretty amazing right? But, it only works for 1 of 5 victory types. So, maybe it's not so great. I don't know. Would this hypothetical city-state be the best in the game? That's kind of what we're debating now. I'm not saying Yerevan is an "instant win button", but out of all the city-states here, it's the closest to that.
Zanzibar [4]