From a rules standpoint, it would remove the spell itself from the stack, but it would not reverse any triggered abilities of the spell.
If you look in the rules, you will see that the definition of countering a spell is to remove the spell from the stack and place it into the graveyard.
Well, you are doing the same thing here, but instead of placing the card into the graveyard, you are putting it back into your hand.
When you counter a Storm card, does that change any effects generated by the triggered ability? No, in fact, it specifically doesn't. That's one of the strengths of Storm is that you have to counter each copy individually.
So yes, which this artifact, you would be able to cast Brain Freeze (for example), put all the copies on the stack, then discard a card to return the original spell to your hand.
A few spells are played (3), you cast Brain Freeze. That's 1+3, but the 1 is removed from the stack. Then cast again, that's 1+4 (because you still cast it, you just removed it from the stack), discard a card. You're opponent now peels 7 cards and you still have Brain Freeze in your hand.
It's a good thing they didn't print a storm card that could draw cards...
