**Core Concept**
The stylohyoid ligament is a fibrous structure connecting the styloid process of the temporal bone to the lesser horn of the hyoid bone. It plays a significant role in the movement and support of the hyoid bone during swallowing and other oral functions. The development of the stylohyoid ligament is closely related to the embryological origins of the branchial arches.

**Why the Correct Answer is Right**
The stylohyoid ligament is derived from the second branchial arch, also known as the hyoid arch. During embryonic development, the second branchial arch gives rise to several important structures, including the stylohyoid ligament, the styloid process of the temporal bone, and the lesser horn of the hyoid bone. The second branchial arch is characterized by the presence of the hyoid muscle, which is responsible for the movement of the hyoid bone. The stylohyoid ligament is a fibrous extension of this muscle, connecting the styloid process to the lesser horn of the hyoid bone.

**Why Each Wrong Option is Incorrect**
**Option A:** The first branchial arch, also known as the mandibular arch, gives rise to structures such as the mandible, the maxilla, and the zygomatic bone. It does not contribute to the development of the stylohyoid ligament.
**Option C:** The first branchial pouch is involved in the development of the middle ear cavity and the Eustachian tube, but it is not related to the formation of the stylohyoid ligament.
**Option D:** The second branchial pouch is involved in the development of the thymus and the inferior parathyroid glands, but it is not related to the formation of the stylohyoid ligament.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that the stylohyoid ligament is a fibrous structure that plays a crucial role in the movement and support of the hyoid bone during swallowing and other oral functions. The development of the stylohyoid ligament is closely related to the embryological origins of the branchial arches, specifically the second branchial arch.

**✓ Correct Answer: B. 2nd branchial arch**