This is something I've been mulling over for awhile. Obviously every character is different, and presents different challenges for the editor and cover artist(s).
The Atom, for example, is tiny. So Julius Schwartz had Gil Kane create covers which presented the Atom in a difficult situation, often precisely because he was so small:
Obviously, no other character could find themselves in those particular situations--menaced by a Venus flytrap, about to be sucked down the drain, or sealed inside a wristwatch. There were many other similar covers, with the Atom stuck to the tire of a speeding vehicle, or strapped to a grenade, or menaced by a cat.
With the Flash, at first the covers emphasized his incredible speed, as in the famed cover to Showcase #4:
He's running so fast that he's actually popping right off the film.
But there are limits to that approach, and so Schwartz and Carmine Infantino began emphasizing his colorful villains:
Indeed, of all of DC's superheroes in the Silver Age, the Flash had by far the most interesting rogues' gallery, while the Atom had hardly any costumed villains. Again, some of this may be due to the nature of the superpowers. The Flash's ability is so strong that it almost requires a lot of trickery, while the Atom's power is much more limited and thus he was more likely to encounter common criminals.
With Green Lantern, Schwartz and Kane went more with puzzle covers. How can this be happening, and how can GL get out of this situation?
With Hawkman, Schwartz and Murphy Anderson tended to emphasize fighting scenes, particularly with the Winged Wonder using some of the medieval weapons and armor he favored:
I'll try to take a look at some of the Marvel covers next.