Lithium is harvested from evaporation pools, not mined. If you're referring to cobalt content, there are several researchers working to remove the need for that element from the production process.
Lithium polymer batteries are highly recyclable, they primarily just have to be unwound, possibly reformed if something structurally changes in the cathode/anode, and the trace elements need to be washed off and processed.
Batteries do not use rare-earth elements (as far as I am aware).
You may be thinking of cobalt, which is expensive and mostly mined in the Congo. Reducing the cobalt content of cathodes is a big priority for the industry:
Also, what do we do with dead solar panels? Is there a recycling program, or research going into such a thing?
I'm so excited for this news and I feel like these are pretty minor cons though.