There are 20 of these cards in paper and they all hail from the original Zendikar block. They are all instants in paper with the subtype of "Trap". The majority of them read something like this "If X happened, you may pay X rather than pay this spell's mana cost." Essentially they are cards that cost less based on something your opponent does.
Here's an exact example:
Permafrost Trap (4)
If an opponent had a green creature enter the battlefield under their control this turn, you may pay rather than pay this spell's mana cost.
Tap up to two target creatures. Those creatures don't untap during their controller's next untap step.
My thought is that instead of Traps having the cost reduction aspect, they would bypass the cast timing limitations of PQ.
So in the case of Permafrost Trap, in PQ it would say "If your opponent had a green creature enter the battlefield this turn, trap. Disable two target creature's until the end of your opponent's next turn."
The trap tooltip would say:
If this card's trap was triggered, you may cast it.
Since this card would be used to lock down attacking creatures right before they attack, it could be made with Flash instead but there are other situations where you would want to cast a paper "instant" when there are no creatures attacking. This is where Trap would come into play. Flash works by being disabled in your hand and your opponent's creatures attacking, trap would simply work by something happening. It could be simplified to have a specific trigger, one that is also already on a paper card:
Mindbreak Trap (4)
Instant — Trap
If an opponent cast three or more spells this turn, you may pay rather than pay this spell's mana cost.
Exile any number of target spells.
Trap could always trigger if your opponent cast three or more cards in one turn. It could be the anti-hand dump mechanic.
Here is an example card:
Drain - U - 8 mana
Each card in your opponent's hand is fully drained of its mana.
It would also have a symbol for trap which means this card will automatically cast once your opponent casts his/her third card in a single turn. Kiora is casting a bunch of gem converters and getting ready to dump her hand and go into a loop? Nope. She casts three and then loses all that excess mana after her third card is cast.
Aura Mutation - WG - 5 mana - Trap evergreen
Destroy an opponent's Enchantment support. Create X 1/1 green Saproling creature tokens, where X is that enchantment's converted mana cost.
This is the type of card I was originally thinking about when I came up with this idea. Instant cast support destruction that doesn't rely on creatures attacking. There are Omniscience decks and Imminent Doom decks out there that will combo off and kill you before you could even use a Flash support destruction to save yourself. In this case you could shut down the combo if you had this card in hand and it had full mana.
Here is my ultimate Omni/ID trump card:
Deconstruct - G - 5 mana - Trap evergreen
This way you don't even need to have mana in the card to shut down Omni and ID. This card doesn't guarantee you permanently shut them down but it stops them from comboing off and killing you out of nowhere. Let me know what you all think.