Final answer:
Kernig's sign is elicited by extending the child's leg at the knee while the hip is flexed at a 90-degree angle, checking for pain or resistance.
Step-by-step explanation:
To elicit Kernig's sign, which is associated with meningitis, the appropriate procedure is to extend the child's leg at the knee and observe for pain or resistance. This is done while the child is lying flat with the hip flexed at a 90-degree angle. If pain and increased resistance occur when attempting to straighten the knee, this indicates a positive Kernig's sign, which could be suggestive of meningitis.