Plantar Fibroma: This is a common condition characterized by a benign growth on the plantar fascia, which can present as a firm, tender lump on the arch of the foot. The fact that massage helps alleviate symptoms is consistent with this diagnosis, as plantar fibromas can be sensitive to pressure but may also feel better with massage or stretching.

Morton's Neuroma: Although typically found between the toes, a neuroma (a benign nerve tumor) could potentially present on the arch, especially if it involves a branch of the nerve that supplies this area. It's tender and can be relieved by massage or pressure reduction.
Ganglion Cyst: A lump under the skin, often on the foot, which is usually harmless but can be painful if it presses on a nearby nerve. Massage might provide temporary relief by reducing pressure on the surrounding tissues.
Bursitis: Inflammation of a bursa (a fluid-filled sac) near the joints can cause a painful lump. The arch of the foot has several bursae that could become inflamed, leading to a tender mass. Massage could help reduce inflammation and alleviate symptoms.

Soft Tissue Sarcoma: Although rare, any unexplained lump, especially one that is firm and growing, should prompt consideration of a soft tissue sarcoma. These are malignant tumors that originate in the soft tissues, including muscles, tendons, fat, and blood vessels. Early detection is crucial for effective treatment.
Infection or Abscess: A lump on the foot could be a sign of an underlying infection, including an abscess. While less likely given the description of the lump being solid and massage helping, it's essential to consider, especially if there are signs of infection like redness, warmth, or pus.
Metastasis: In rare cases, a lump on the foot could be a metastasis from a cancer elsewhere in the body. This would be extremely uncommon but is a "do not miss" diagnosis due to its serious implications.

Lipoma: A benign tumor composed of fat tissue. While lipomas can occur almost anywhere, they are less common on the foot, especially on the arch.
Glomus Tumor: A rare, benign tumor of the glomus body, usually found under the fingernail but can occur on the foot. It's typically very painful and sensitive to temperature changes.
Plantar Wart: Caused by the human papillomavirus (HPV), plantar warts can appear as hard, grainy growths on the bottom of the foot. They might be tender but are usually not described as a solid mass.