Step 1: Think about the three germ-layer contributions to a single pharyngeal arch. The outer covering is ectoderm, the inner lining is endoderm, and the central mass is mesoderm. None of these three on its own supplies the arch nerves.
Step 2: A fourth, migratory population - the neural crest - seeds each arch. Crest cells originate at the dorsal lip of the neural folds, then stream ventrolaterally into the arches where they generate peripheral neurons, Schwann cells and connective tissue. So the answer for the nerve supply is the neural crest.
Step 3: Cross out the distractors. Mesoderm gives rise to the arch musculature and endothelium; endoderm builds pouch organs such as the thymus and parathyroids; neuroectoderm (the neural tube proper) makes the brain and spinal cord, not the peripheral arch nerves.
\[\boxed{\text{Neural crest}}\]