In the new 2021 stimulus pkg, unemployment benefit payments up to $10,200 will not be taxable. Can I file an amended return to include this?

That tax law has not yet been signed into law by the President.   When that happens then the IRS will need to change the tax forms, the programmers have to change the software, and then the IRS has to approve of their software.   That will take time---we do not know yet when all of those changes will be accomplished.  As soon as more is known, TurboTax will provide information.  The software changes will take weeks or even several months.

I just received a direct deposit into my checking account with the explanation :

Is this the refund for the taxes I paid in 2020 on my unemployment benefits?  If so, will I ever receive any paperwork explaining this?

Yes it could be.  That is the description for IRS refunds.  Don't know if they are sending out a letter explaining the calculation.