What are my rights if my child swallowed a medicine because there was no child proof lid?

This area of law is called products liability.  You may be able to file a lawsuit for negligence against the manufacturer and the seller (store where you purchased the product) of the product.  In order to prove negligence, you would have to prove that the manufacturer breached a duty of due care (that degree of care that a reasonable manufacturer would have exercised to prevent foreseeable harm) by failing to have a child proof cap and the poison control number on the bottle.  You would also have to prove that the breach of this duty of care was the actual and proximate cause of your child's injury.  Actual cause means but for the child proof cap and/or warnings and/or poison control number would your child have been injured?  If the answer is no, you have established actual cause.  Proximate cause means were there any unforeseeable intervening events that would relieve the manufacturer of liability?  If the answer is no, you have established proximate cause and the manufacturer is liable for negligence.

Under products liability, your lawsuit should include a separate cause of action for strict liability.  Strict liability means that the manufacturer is liable even if the manufacturer exercised due care.

As for the seller (store), in products liability cases, the seller of the product is liable even if the seller did not know or could not have discovered that the product was defective.  Your lawsuit against the manufacturer of the product would also name the seller as a separate defendant in your claims for negligence and strict liability.

Your damages (the monetary amount to be recovered) would be determined by the injury to your child including medical bills and compensation for pain and suffering.  Compensation for pain and suffering is an amount in addition to the medical bills and is determined by the medical reports which document the nature and extent of the injury.   Fortunately, your daughter's injury was not serious which would result in a reduced amount of monetary damages. 

The defendants (manufacturer and store) could assert the defense of assumption of the risk considering that the pills were within your daughter's reach and she was able to climb on the counter and microwave to grab the bottle.  The defendants might argue that parental supervision should have prevented this or that the bottle should have been kept out of your child's reach.  Your counterargument is that the child's injury is foreseeable because of the manufacturer's failure to have a child proof cap on the bottle and any warnings or poison control phone number on the bottle.