Question: 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.

Answer: 
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.

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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...

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> 1. The spatula edge should stay on the pan so that the rolling of the egg off the spatula is "[easy](http://www.ehow.com/how_2051043_fry-egg-over-easy.html#page=0)".
> 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](http://ask.metafilter.com/6002/eggs-over-I-can-understand-over-easy-though-is-hard).
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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.