To me, "over easy" seems to refer to the pan, because I think of a pan sitting atop an easy flame. "Over hard" makes me think it refers to the egg, because the liquid becomes solid. Then again, maybe it's something else altogether.
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...
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.)
Most likely, easy refers to going easy on the temperature and duration, and hard on the condition of the egg, as you suspect. We can play hard and work hard, but we don't really speak about cooking hard. The egg yolk gets solidifies atop even an easy flame, given enough time. And of course, easy doesn't refer to the liquid condition of the yolk.
These terms refer to the egg, namely to its degree of cooked-ness: over-easy is cooked on both sides but yolk is runny; over-medium is less runny than over-easy, and over-hard is when both the yolk and white is cooked all the way through.