Is that a separate objection to the one he has against relativity or did the journalist just mix things up? Because Einstein did little or no work in the Big Bang theory, that's due to Lemâitre, Friedmann, Robertson, Walker, and a few hundred others. But anyway, the thing is: okay, after quark-gluon plasma cools down we have primordial nucleosynthesis that fiils the universe with hydrogen, and after a few thousand years we have stellar formation. We know we do have stellar formation because we've seen galaxies that are like only 300,000 years older than the universe.
So we've established that there were very old stars at the beginning of the universe. Now, how long could they last? Let's see how long stars last today. A red dwarf (like Barnard's star) can live for a few trillion years (so it's conceivable some of the first stars are still around), a yellow dwarf like the Sun or Tau Ceti lives for a few billion years, while very massive giants (like Rigel or Betelgeuse) last only few million years. Now let's see, okay, at first we only have hydrogen, but the first stars fuse the hell out of it, forming helium, carbon and oxygen. Then the heavier stars (doesn't need to be
too heavy, twice the Sun's mass is more than enough) begin to fuse iron, and go supernova (forming even more heavier elements) all over the place. If we look out the window, we see the universe today being
littered with supernova remnants. There's enough time since the beginning of the universe for stars to form, explode, reform and explode again,
a thousand times over.
Then we have the meta-arguments: if the Big Bang theory truly couldn't explain how heavy elements formed (btw: the Earth is made mostly of iron, not carbon), even after models of cosmological evolution having been simulated to death, people would kind of
hear about it. Nah, the Big Bang theory has a few problems as we understand it (mostly, we have no way to explain how there's so much matter and barely any antimatter), but this is not one of them.
A penny for every time a 12 year old claims to have a "theory"... overzealous parents really need to stop flaunting their damn kids so much, maybe they'll stop thinking they're smarter than the generations of brilliant and more experienced scientists that came before. I mean, of course it's impressive that he shows a decent (not great, decent) understanding of univariate calculus at 12, and he may or may not have an interesting future in science, but science is a marathon, not a sprint. Of course there are some kid geniuses. Julian Schwinger got a PhD when he was 18. But Albert Einstein, for example, received his when he was 26. Science has a lot more to do with having the creativity to ask the right questions than the ability to learn things that already exist.
tl;dr: the kid knows some calculus, that's nice. Still, that doesn't make him any more qualified to make "theories" about cosmology and gravitation than a random sophomore engineering student.
Edit: found a nice and very readable Hertzprung-Russell diagram relating star lifetimes with their luminosity and spectral type. Pretty nice. I want a poster of it. Click for larger, screen breaking version.