My opponent played Nahiri's Wrath and I played Convolute. As an additional cost to cast he had to discard a card. He chose a card with Madness Alchemist's Greeting. What happens to the card that was discarded to pay the extra cost when Nahiri's Wrath is countered? Is it able to be played by madness cost, go to the graveyard, or go back in the hand?
The card was discarded as part of the cost to cast Nahiri's Wrath, when this happens, it's Madness ability trigger. So it is discarded into exile, from there when the Madness trigger resolves, your opponent can cast Alchemist's Greeting for it's Madness cost if they so chooce.
This is all independent of the original spell, Nahiri's Wrath. So even if Nahiri's Wrath is countered, your opponent can still cast the Madness card if they choose.
Your description is a bit out of order. Your opponent pays the additional cost of Nahiri's Wrath when the spell is played, before you can even declare you're countering it with Convolute.
So first, as your opponent plays Nahiri's Wrath, he declares the spell's targets, and pay the spell's cost - including its mana cost and its additional cost of discarding cards. Alchemist's Greeting is discarded into exile and its Madness trigger goes to the stack.
Only then, after your opponent has passed to you, you can cast Convolute in response. Countering the spell doesn't refund its cost (so the discarded cards won't be returned to hand, just like the 2R mana spent doesn't return to his mana pool), and has no effect on the Madness trigger.
So yes, your opponent will be able to cast Alchemist's Greeting for its Madness cost, if he wants.
This is all independent of the original spell, Nahiri's Wrath. So even if Nahiri's Wrath is countered, your opponent can still cast the Madness card if they choose.
So first, as your opponent plays Nahiri's Wrath, he declares the spell's targets, and pay the spell's cost - including its mana cost and its additional cost of discarding cards. Alchemist's Greeting is discarded into exile and its Madness trigger goes to the stack.
Only then, after your opponent has passed to you, you can cast Convolute in response. Countering the spell doesn't refund its cost (so the discarded cards won't be returned to hand, just like the 2R mana spent doesn't return to his mana pool), and has no effect on the Madness trigger.
So yes, your opponent will be able to cast Alchemist's Greeting for its Madness cost, if he wants.