The big issue with this sort of thing is that, somehow, all different parts of the superhero's body need to be affected at once, and with synergy. If half of the hero's muscle fibers are made stronger, and half are made weaker, they gained nothing. And then there's the whole issue of all the unlucky mutations that kill the cells outright (and there's far more of those than there are mutations for telepathy!). There's no way a simple radiation accident (or similar) could do what we want on its own. However, what if the seeds for this ability are millions of years old? Perhaps instead of having the radiation cause the mutation, maybe all it does is trigger it, and some complex internal system does the rest.
The current favored hypothesis for the evolution of species is called punctuated equilibrium. In this system, species spend most of their time in a sort of equilibrium where few changes occur. However, every now and then, they undergo a period of rapid change. This rapidity is used to explain the lack of transition species that we can find in the fossil record. Now the real scientific theory this occurs over hundreds of generations, but what if it actually occurred in just one?
What if our bodies ran in two modes, which I will label "user mode" and "kernel mode" because there's enough similarity between this idea and the way modern computers handle privileges. The key difference between the modes is that "user mode" is not allowed to alter the DNA itself, except through well understood channels such as mitosis and meosis.
Kernel mode would be invoked via a quorum. If enough cells signal that we need to unlock "the hidden potential" of the individual, they'd vote for it with hormones. If enough hormones are emitted, it's like the bat signal flashing over the sky, and the kernel mode capacities of our genome are woken up. They call one final vote (which is easier now, because they have an "elected official"), and if the vote is sufficient, kernel mode is activated.
In kernel mode, the DNA of the individual is open for reading and writing. This would be very powerful in radiation environments because it would act like a massive RAID of harddrives. Every bit of damage that gets caused has a few million replicas elsewhere in the body. This could counteract the otherwise certainly-fatal doses of radiation poisoning.
Of course, if you were going to have this sort of power hidden in your genome, you'd not only want to be able to counteract the radiation, you'd want to rise above it. You'd want to write new genes to rise above the environment. These genes, of course, could be passed on if you chose to rewrite your germline DNA. Your brain, on its own, would be insufficient to do this but what if it had help? What if, in some of the parts of our genome which appear to be unused likes a sort of grimoire of abilities the genome knows how to do. This sort of knowledge could then be used by the lower brain to try to identify mutations to fix their environment.
I say lower brain for three reasons. The first is that it's older... this sort of crazy pseudo-science would have to be born quite early if it were responsible for punctuated equilibrium. Perhaps the kernel mode of our bodies doesn't yet trust this new fangled cerebral cortex. The second reason is that, in the stories, people tend not to have full control over what mutations occur. If the cerebral cortex was involved, they would. Finally, I'd like to point out that most individuals going through this process are... well.. screaming in agony and in no condition to make life decisions.
I can't make any claim that this is how it works in real life, because I'd almost certainly be wrong. However, this seems like a reasonable way to bundle up just that much otherwise unbelievable power into a momentary dose of radiation. The radiation isn't doing the mutation, it's merely stressing the body enough to break a seal and unleash the next higher level of power the body has.