Referred pain from a dental issue: The description of "toothache pain" suggests a possible dental origin, and pain can sometimes be referred to other areas, including the hands, though this is less common. The most straightforward explanation would be a misinterpretation of the location of pain, with the actual source being dental.

Neuropathic pain: Conditions affecting the nerves, such as neuropathy, could cause pain in the hand that might be described as similar to a toothache due to its sharp, stabbing nature.
Arthritis or osteoarthritis: Inflammation or degeneration of the joints in the hand could cause pain that might be confused with a toothache, especially if the pain is localized and sharp.
Tendinitis or ligament sprain: Injuries to the tendons or ligaments in the hand could cause localized pain that might be described in terms of its severity and character similar to a toothache.

Complex Regional Pain Syndrome (CRPS): A condition characterized by intense burning or aching pain in the hands or feet after an injury or trauma. It's crucial to identify CRPS early due to its potential for severe, long-term disability.
Infection or abscess: Though less likely, an infection in the hand could cause severe pain. Given the potential for serious complications, including sepsis, it's essential to consider and rule out infection.
Vascular occlusion: Conditions like Raynaud's phenomenon or a vascular occlusion could cause pain in the hand, and while less common, missing a vascular issue could have significant consequences.

Syringomyelia: A rare condition where a cyst forms within the spinal cord, potentially causing pain, weakness, and stiffness in the hands.
Peripheral nerve tumors: Benign or malignant tumors affecting the nerves in the hand could cause localized pain.
Erythromelalgia: A rare disorder characterized by burning pain, heat, and redness in the hands and feet, which could potentially be described as similar to a toothache in severity.