What are my mother’s rights if was told this afternoon by her doctor that while in the hospital a contaminated needle from another patient was used on her?

Medical malpractice is negligence.  Negligence is the failure to exercise due care (that degree of care in this case that a reasonable hospital would have exercised under the same or similar circumstances to prevent foreseeable harm).

Your mother could sue the hospital for negligence for using the contaminated needle.  Prior to filing a lawsuit, it may be possible to settle the case with the hospital's insurance carrier.  Your mother's personal injury claim filed with the hospital's insurance carrier should include her medical bills (resulting from the contaminated needle), medical reports and if applicable documentation of any wage loss.  The claim should be filed wihen your mother is either released by the doctor upon completion of treatment or is declared by the doctor to be permanent and stationary which means having reached a point where no further improvement is anticipated.  Compensation for the medical bills is straight reimbursement.  The medical reports will document the nature and extent of the injury and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  Compensation for wage loss is straight reimbursement.

If your mother is dissatisfied with settlement offers from the hospital's insurance carrier, she should reject the settlement offers and file a lawsuit for negligence against the hospital.  If the case is settled with the hospital's insurance carrier, NO lawsuit is filed.  If the case is NOT settled with the hospital's insurance carrier, your mother will need to file her lawsuit for negligence prior to the expiration of the applicable statute of limitations or she will lose her rights forever in the matter.