Well, meteorite minerals like iridium and all aren't really found on Earth in appreciable quantities.
What you're looking for are exotic atoms. These certainly exist, but are too unstable. And, for certain exotic atoms like onia, atomic number isn't even defined.
The binding forces cannot be different since the coupling constants are...well... constant (not sure what string theory says about this--but the Standard Model keeps them constant).
One thing that I can think of are "satoms" (atominos?), made of sprotons, sneutrons, and selectrons. Or maybe some other superparticles. Supersymmetry predicts that each particle has a superpartner. These ought to exist in our universe, but we haven't detected any yet. They are a candidate for dark matter though.
I'm not too sure of how superparticle stability works, though. Seems like only one of them is stable. We could make an atom out of that, I guess. But it's electrically neutral, and probably very light. So there may or may not be sufficient force holding it together.
As @annav said, what you may be looking for is a new alloy or something. THis will be an "exotic metal", but will still be made of normal atoms.