Why are eggs fried (over-)easy while boiled soft? I looked for etymological clues on the phrase "over-easy", but could find one. In both frying and boiling, fully cooked eggs are hard in the sense of stiffness, not of difficulty, so I can't understand where the easiness is coming from.
In both frying and boiling, fully cooked eggs are hard in the sense of stiffness . . .
No, this is not true.
When frying eggs, easy and hard are flat adverbs referring to how gently (or not) you flip the eggs.
"I want my eggs over-easy."
→ "Be gentle with my eggs. Flip them easy.
In other words, flip the eggs in such a way that the egg yolks don't break. (Whether or not the egg yolks ooze when cut into after the eggs have been served depends on how long they have been fried after having been flipped. Normally, an oozing yolk is the intended result—but it doesn't have to be.)
"I want my eggs over-hard."
→ "Be aggressive with my eggs. Flip them hard.
In other words, flip the eggs in such a way that the egg yolks do break. (There should never be runny yolks when the eggs are served. The yolks will have spread out into the whites when cooked.) Sometimes, people will deliberately break the egg yolks before the eggs are flipped in order to accomplish the same effect. (Breaking an egg on flipping it is not actually simple.)
The description is one that describes the way in which they are flipped over. (Hence, the over- prefix.)
Note that a sunny side up egg is one that is fried but never flipped at all.
When boiling eggs, easy and hard are adjectives that describe the texture of the result. As you say, the words describe the stiffness (generally of the yolk).
Boiled eggs are neither over-easy nor over-hard. They are not flipped over at all. They are simply soft or hard as a result of the cooking time.