What to do if an offer of employment was rescinded because of incorrect information reported on background check?

You can sue the background check company for negligence.  Negligence is the failure to exercise due care (that degree of care that in this case a reasonable background check company would have exercised under the same or similar circumstances to prevent foreseeable harm). 

In order to prove negligence, you will need to prove duty (duty of care mentioned above), breach of duty (failure to exercise due care in performing the background check), actual cause, proximate cause, and damages.  Actual cause means but for the background check would the job offer have been rescinded?  If the answer is no, you have established actual cause.  Proximate cause means were there any unforeseeable, intervening events which would relieve the background check company of liability?  If the answer is no, you have established proximate cause.  Damages means the amount you are seeking to recover in your lawsuit.  Damages would be based on lost income from the job offer that was rescinded due to the background check.  You will need to mitigate (minimize) damages by making reasonable efforts to find a comparable job.  If you find another job that pays less, your damages would be the difference in income between the job offer that was rescinded and the job for which you are hired.  If you fail to mitigate damages, your damages will be reduced accordingly.

You will need to file your lawsuit for negligence against the background check company prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.