Superman being solar-powered is a prime example. Originally all Kryptonians had the powers he had (which were a lot less/weaker than he had later), even while on Krypton, and he had powers from the moment he arrived on Earth. Eventually in the Silver Age that changed having power on Earth due to the combination of yellow sun and lower gravity, and he lost powers when placed in an environment replicating Krypton, such as visiting Kandor. The reboot in the 1980s had him initially "normal" when he arrived on Earth and gradually gaining power due to exposure to the sun, and that's stayed pretty much the same through the last 30 years.
Green Lantern (Hal Jordan and later) and the other Lanterns initially could not affect yellow objects or other yellow things because of something technobabble reason, and the primary characteristic of being a Lantern was being fearless. Green Lantern: Rebirth in 2005 revealed the yellow issue was due Parallax being trapped and revealed the emotional spectrum. Green Lanterns weren't powered due to lack of fear, they were powered by Will, the Star Sapphires by love, the Sinestro Corps by fear, and so on. That doesn't seem to be going away any time soon.
Spider-Man went with the radioactive spider, but over the last while that's quietly been shuffled aside for a genetically-engineered spider, or at least one where the radiation induced a mutation in the spider, not the radiation itself transferring to Peter.
Speedsters in the DC Universe provide another example as they originally had different explanations for going fast. Barry Allen was doused in chemicals and struck by lightning. Jay Garrick inhaled heavy water vapour. Johnny Quick (and later Jesse Quick) recited a mathematical formula. Black Racer simply had power because he was a New God. Edward Clariss (the Rival) created a drug that temporarily granted super-speed. Eobard Thawne, the first Reverse-Flash, used a tachyon device. Hunter Zolomon, the second Reverse-Flash, gained his speed from an accident involving attempted time-travel. Eventually, the origin of their powers was retconned into being the Speed Force, giving them a common source, and that's now an accepted, standard part of the Flash mythology.
Also in the DC Universe is the concept of the "metagene", that about 12% of humans, in a sufficiently traumatic situation, can spontaneously develop some kind of power in response to save their lives. That was first introduced in the 1988-89 "Invasion!" crossover and has been in DC comics ever since as an explanation for the source of powers of assorted characters, good and bad.