Rare earth minerals are not rare, they are everywhere, called rare earth because they don't come in veins or seams but are scattered about and much earth must be moved and refined to gather them up.
Exactly right. There are a number of places in the US, Europe, Australia and other places that could each produce 20% or more of the world's needs for rare earths. In fact, a lot of waste from other mining operations are lanthanides (chemist's term for rare earths) - remember that toxic red sludge that burst out of the holding tank in Hungary a few weeks back? Full of lanthanides among other things. The problem is really the extraction and separation of the lanthanides once you've mined them. Because of their chemical similarity there is no easy way to do this. It is a horribly labor intensive, dangerous and polluting process right now - exactly the sort of thing the Chinese are willing to do, and reasonably good at.
Fortunately, since China showed their hand and got people thinking, there are now a bunch of smart people looking for better ways to handle the extraction and separation of the lanthanides. So, even if China does hold out on the rest of the world, the rest of the world can take up the slack. Sure, there will be a few bumps for the first few years but ultimately the end results might actually be a good thing.
Just a Shooter (who paid attention in Chemistry class)