Step 1: Branchial cleft cysts arise from incomplete obliteration of the embryonic branchial apparatus and surface as a smooth, painless mass in the lateral neck of a young patient.
Step 2: Anatomically they sit in the anterior triangle, hugging the upper one third and the anterior margin of the sternocleidomastoid muscle.
Step 3: Lesions tied to the posterior border, such as cystic hygroma, and reference points like the digastric or omohyoid muscles do not describe the usual branchial cyst location.
Step 4: The most frequent site is therefore the anterior border of the sternocleidomastoid.\[\boxed{\text{Anterior border of SCM}}\]