Lymphadenitis or Reactive Lymph Node: This is the most likely diagnosis given the location under the arm pit and the fact that the lump has been present for months without changing size. The tenderness could be indicative of an inflammatory process, and the absence of other symptoms such as discharge or itchiness suggests a benign cause.

Lipoma: A benign tumor made of fat tissue, which can occur under the arm pit and may feel tender, especially if it's pressing on surrounding tissues.
Cyst (e.g., Epidermoid or Sebaceous Cyst): These are benign, fluid-filled lumps that can become tender if infected or inflamed. They are common in areas with sebaceous glands, including under the arms.
Hidradenitis Suppurativa: Although this condition often presents with multiple lesions and can be itchy, early stages might not have all these symptoms. It's an inflammatory condition affecting the apocrine glands.

Lymphoma: Although less common, lymphoma can present as a painless or tender lymph node enlargement. Given the potential severity of this diagnosis, it's crucial to consider it, even if the likelihood is low.
Metastatic Disease: In rare cases, a lump under the arm could be a sign of metastasis from another cancer. This would be extremely rare in a 16-year-old but should be considered to ensure no serious condition is overlooked.
Tuberculosis (TB): TB can cause lymphadenopathy (enlargement of the lymph nodes), which might be tender. Although less common in some regions, TB is a critical diagnosis not to miss due to its public health implications and the need for specific treatment.

Soft Tissue Sarcoma: These are rare tumors of the soft tissues (such as fat, muscle, or blood vessels) and can occur under the arm. They are more likely to be painless but can cause discomfort if they press on surrounding structures.
Cat Scratch Disease: Caused by Bartonella henselae, this infection can lead to lymphadenopathy, typically after a cat scratch or bite. The lymph nodes may be tender and can take months to resolve.
Brucellosis: A zoonotic infection that can cause lymphadenopathy among other symptoms. It's rare and usually related to exposure to infected animals or their products.