In the context of D&Dnext, it has to be something distinctive about a great weapon fighter, because they're the ones who get damage on a miss. (Well, also the 20th level rogue via Ace in the Hole, but for whatever reason no one seems to care so much about that one.)
Let's take a look at the ability.
Ace in the Hole
By 20th level, you have an uncanny knack for succeeding when you need to. If your attack misses a target, you can turn the miss into a hit against that target. If you can take actions and fail an ability check or a saving throw, you can treat the d20 roll as a 20. Once you’ve used this feature, you regain its use only after you complete a short rest or a long rest.
At first blush I would say the reasons we do not care are the following:
1. It is a 20th level ability. not a first level ability like Great Weapon Fighting.
2. "If your attack misses, you can turn it into a hit." Is not the same as "When you miss a target with a melee weapon, the target still takes damage."
3. It works only once per fight, as you need a short rest or long rest after its use. Therefore it is limited in use, not used on EVERY round of every fight all day long.
Again, that is just a first blush. If the fighter version had all these restrictions I will withdraw my objection. Remember, must be limited use, 20th level, and conditional that it MIGHT still hit, not an auto-hit. Also no longer bypasses DR, avoids poison, etc..
I am also sure there are other reasons why it makes more sense.
Oh well then, to my mind, a Pixie who successfully avoided a weapon swing might still lose HP from...
...straining a wing.
...feeling a bit of mortal fear.
...losing some morale.
...losing a bit of luck. (Still don't know what that would mean.)
...getting disoriented from the maneuver used.
...getting fatigued from the maneuver/effort required to avoid the blow.
...probably a few other things someone more creative can come up with.
All of which, AFAICT, would be compatible with the definition of HP presented in the "How to Play" document.
A fighter who (successfully or not) swings his sword might similarly lose HP then from...
...straining his arm.
...feeling a bit of mortal fear (the pixie in question is immortal.. so immortal fear then I guess)
...losing some morale.
...losing a bit of luck. (And I am right with you in not knowing what that would mean.)
...getting disoriented from the maneuver used. Over correcting from a failed attack perhaps.
...getting fatigured from the maneuver/effort required to swing the sword.
...probably a few other things that someone more creative can come up with.
Except there is no reason why the fighter would have these difficulties. And yet it would make more sense that he would tire out by wielding a greatsword than an immortal pixie would by moving a few inches out of the way.
Also, no one except the fighter with a greatsword can cause the pixie to strain her wing. Not the guy with two weapons, nor the relentless barbarian, or any other class or combination EXCEPT the greatsword fighter ..on a miss.