Once per (short or long?) rest you may get an additional reaction
That's the most I'd see this sort of thing being. Even then would many people take it?
EDIT: Coming to realize that a reaction turn is essentially an extra turn that can be used by a character like a rogue to do even more sneak attack damage on top of his previous reaction, this might be very much questionable. For a swashbuckler archetype this would definitely be something. For spellcasters and throwing an additional reaction to use something like counter-spell is another.
How strong would a feat for getting an extra reaction per round be?
Too strong for a feat.
The Forum Infestation (TM)
That's the most I'd see this sort of thing being. Even then would many people take it?
EDIT: Coming to realize that a reaction turn is essentially an extra turn that can be used by a character like a rogue to do even more sneak attack damage on top of his previous reaction, this might be very much questionable. For a swashbuckler archetype this would definitely be something. For spellcasters and throwing an additional reaction to use something like counter-spell is another.