Step 1: Frame the question by tracing each germ layer contribution in a pharyngeal arch. The arch has three coverings and a core: outer ectoderm, inner endoderm, and a central mesodermal mass that is heavily invaded by migrating cells.
Step 2: The decisive migrating population is the cranial neural crest. These cells detach from the dorsal neural tube margin and travel into the arches, where they are called ectomesenchyme and supply connective tissue, cartilage, bone and, critically, the neural elements.
Step 3: Because the arch nerves grow from this crest-derived neural population (peripheral neurons and glia), the embryonic source of the branchial arch nerves is the neural crest, so option c is selected.
Step 4: Eliminating alternatives: mesoderm is responsible for arch musculature and aortic arch vessels; endoderm lines the pharyngeal pouches; neuroectoderm builds the brain and spinal cord, not the peripheral arch nerves. The single best match remains the neural crest.
\[\boxed{\text{Neural crest}}\]