Can a person who received a defective gift that caused them harm sue the manufacturer?

This area of law is products liability.  You can sue the manufacturer and the seller (store) where the item was purchased.  Your lawsuit should have two causes of action (claims).  One cause of action is negligence and the second cause of action is strict liability.

Negligence is based on the exercise of due care which is that degree of care that (in this case) a reasonable manufacturer would have exercised to prevent foreseeable harm.  Products liability can be either a manufacturing defect or a design defect.

In addition to the manufacturer, the seller of the product is liable even if the seller could not have determined that the product was defective.

Strict liability imposes liability whether or not there was due care.

If you were injured by the defective product, obtain your medical bills and medical reports when you are released by the doctor.  The medical reports will document the nature and extent of your injuries and will be used to determine the amount of compensation you receive for pain and suffering.  Compensation for pain and suffering is an amount in addition to the medical bills.  Your claim should also include any wage loss due to your injury.  Compensation for the medical bills and wage loss is straight reimbursement.

A defense which the manufacturer and seller could assert in both negligence and strict liability claims is assumption of the risk.  Assumption of the risk means that you recognized and understood the danger and voluntarily chose to encounter it.  For example, if you discovered the product was defective and used it anyway resulting in injury or you disregarded warnings on the product and improperly used the product.

You will need to file your lawsuit prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.