Welcome to puberty! Here's your ticket to the adolescent world and all the wonders that come with it: your growth spurts, voice changes, hormone fluxes, and most importantly....your super-special powers/abilities that you've lived the past thirteen years of your life knowing nothing about! And would you look at what it's wrapped in: an impressive secret destiny that your guardians (who probably aren't your parents) have kept hidden from you your entire life. If you're lucky, it'll come with a special heirloom/talisman that only you can use! Well, that's it, everything you need for your journey into adulthood. Have fun and watch out for your completely evil nemesis!
This seems to be the course of many fantasy books, especially "High Fantasy" books. Now don't worry, this isn't a rant about plot-device originality, that's another post for another time 
But honestly, why is it that most character abilities develop mostly in the teen years without the character ever knowing about them before that? I actually think I might have the answer(s) to that question, (on a very theoretical level of course).First off, there's the pretty weighty of those who've gone before. I feel it would be safer to say that this influence is largely an unconscious phenomenon, but it would be a bit silly to deny its existence. The books we read and movies we watch feed our imaginations and shape the kinds of stories we craft, even if we're not always aware of it. So maybe one of the reasons characters' powers pop up at puberty is because that's simply when powers pop up in many stories
But that doesn't really answer the question. After all, what got people started doing this in the first place? That's a bit too historical a question for me to answer it. I mean, I'd have to research and hunt down the very first fantasy story in which a young adolescent develops special powers. And even then I'd still not be able to see the reason behind it. I'd have to ask each and every author personally about why they did it.
However, even though I can't authoritatively answer the question, I have my own suspicion about the allure of adolescents with powers: the struggle to come to terms with new powers/abilities symbolizes that everyone goes through as he/she matures into an adult. Like all good symbols, it allows the author to communicate the struggle to us in a way that our rational minds isn't quite aware of or even able to express by itself. And just as it can be subconsciously received, I think it can be subconsciously employed. I seriously doubt anyone sits down and says, "I'm going to use Aloron's struggle to control his new-found ability to breathe lightning as a metaphor of his struggle to mature into an adult." That doesn't negate its value as a metaphor. And if the appeal to readers was unconscious, then I think it not illogical to infer that its appeal to authors could be unconscious as well.
However, this explanation of the tendency to throw powers to teenagers doesn't justify its thoughtless use by authors.
For instance, consider this: every change that takes place during puberty has had its foundations laid from the beginning of the child's life. Take the growth spurt, it gives extra growth to arms and legs that the child has had and been able to use for their entire lives. I used a safe example, but just about all the other changes are changes of already-functioning features of the person. So why not have special abilities work the same way? The abilities could be there all along in a weakened state with the character aware of them from the start and they grow and develop along with the rest of the child, and their development accelerates along with all other development in the child during puberty.
Also, the changes of puberty are far from instantaneous, the same should probably be true of your characters' abilities. In my personal opinion it's a bit too easy when a character's abilities manifest at full strength almost overnight. Most authors that I've seen, handle that pretty well with character abilities, making them difficult abilities to master that take practice and training. However, most books only cover a span of a few months, and that still seems a bit too easy to me. I mean, puberty lasts for a few years, couldn't power development do the same? Yes, some of the changes are few within a few months, but I think our readers could survive if a character's abilities take more than just a few months to fully develop. Also, in some books it's the characters' control/use of their abilities that take time to develop more than the abilities themselves.
From a worldbuilding perspective, if your characters' abilities are common to their race, then surely that race would have incorporated the development of those abilities into how their culture raises children. The Sci-fi Lorien Legacies series (I Am Number Four and The Power of Six) handles that pretty well. The characters are the last of their race along with their guardians who help them with the development of their gifts. Most of the guardians wind up being killed by the bad guys, but the reader is still given brief glimpses into how the race implemented power-development into their culture. Also, the characters have spent the past seven years of their life looking forward to their powers manifesting around age 13 or so (sound like another point I recently made?) Similarly, even if your characters are the last of their kind, you can still at least make it look like their race had a culture before it died off (another post for a later date
).Another thought is to have your characters develop powers that aren't tied up with some secret destiny. Or maybe like the Lorien Legacies characters, your characters grow up knowing about their destiny.
Or what if your characters with special abilities is called to fulfill a destiny prior to puberty?
Just some thoughts.
That is an amazing idea.