Are there any natural remedies for a toddler's pink-eye?

Relief, not cure. Most pink eye in childhood and infancy is due to viral conjunctivitis. You can relieve the symptoms with warm soaks and bland eyedrops to clear out secretions. Sometimes there is bacterial infection - look for swelling, pus, crusting. If these are present see your ophthalmologist. Home remedies can relieve symptoms but not cure the problem.