Eggs fried/scrambled "over easy" or "over hard" — refers to the pan, the egg, or something else?

There's not really such a thing as over easy scrambled eggs. It originates with fried eggs, where it means turn them over gently (easy, carefully) and cook for for a few seconds more before serving, so the white is fully set (not "snotty"). That's gently because you don't want to break the yolk.

The opposite to over easy is sunny side up (i.e. - don't do that turning over).

EDIT: Because over easy is so well-known in the above context, and because there are no other standard terms to describe how you want your eggs (unlike, say, rare, medium, bloody, well-done for steaks), people do sometimes extrapolate variants such as over hard, or apply over easy to scrambled eggs. They're easily understood, but such usages aren't really standard terminology.

EDIT2: It never occurred to me anyone would propose an alternative origin for over easy. Having scoured the Internet, I don't see anything looking remotely like an "authoritative" etymological reference - so unless someone else does, all I can offer is a couple of links supporting what I think...

1. The spatula edge should stay on the pan so that the rolling of the egg off the spatula is "easy".
2. ...in order to faciliate lightly cooking the yolk, you would have to flip the eggs over, and to prevent the yolk from breaking (and rendering the eggs "cooked hard") you have to flip them over easy.

I'm pretty sure that "over hard" came first: it's diner jargon that the eggs should be flipped over and cooked until they are hard. Now if you want it flipped but not hard, well the opposite of hard is easy, right? (Sure, you could say soft or runny, but diner slang is often supposed to be funny, a bit of an in-joke, and opaque.)