List of blue counterspells that can be played for free?
There are 5 Blue counterspells that can be cast for alternate costs:


Daze
Disrupting Shoal
Foil
Force of Will
Thwart


I found these with the following search on Scryfall:


  o:"rather than pay ~'s mana cost" o:counter c:u


This searches for spells with alternative costs (first clause) with "counter" in the text (second clause) that are blue (third clause).

In addition, Mental Misstep only costs {UP}, so you can pay life instead of mana.

If you also want to count spells with mana that needs to be paid later, Pact of Negation has a mana cost of {0}, but it also says


  At the beginning of your next upkeep, pay {3}{U}{U}. If you don't, you lose the game.


Besides that, there are 3 spells that can be used to effectively counter a spell that can be cast without paying any mana:


Commandeer lets you take control of a spell and change its targets. You can choose a target that is (more) beneficial to you, or, if the other spell is a counterspell, you can redirect it to target Commandeer so that it is countered for having an illegal target after Commandeer leaves the stack.
Mindbreak Trap lets you exile any number of spells, but you can only cast it for free if your opponent has cast at least 3 spells this turn.
Misdirection also lets you choose new targets for a spell. Of course, it is only useful for dealing with targeted spells, and you can use it to deal with counterspells the same way you can use Commandeer.