Adrenal Insufficiency: This condition, including Addison's disease, can lead to low sodium (hyponatremia) due to decreased aldosterone production, which affects sodium retention. High potassium (hyperkalemia) is also a result of low aldosterone, as aldosterone stimulates potassium excretion. Low calcium (hypocalcemia) can occur due to the decreased production of vitamin D in some cases of adrenal insufficiency. The combination of these electrolyte imbalances, along with clinical symptoms such as fatigue, weight loss, and skin changes, makes adrenal insufficiency a strong candidate.

Renal Failure: Both acute and chronic kidney failure can lead to electrolyte imbalances. The kidneys play a crucial role in maintaining sodium, potassium, and calcium balance. In renal failure, the ability to excrete potassium is impaired, leading to hyperkalemia. Sodium levels can be low due to impaired reabsorption or excessive loss. Calcium levels can be low due to impaired activation of vitamin D and hyperphosphatemia, which can lead to secondary hyperparathyroidism.
Hypoparathyroidism: This condition is characterized by the underproduction of parathyroid hormone (PTH), which is crucial for calcium homeostasis. Low PTH leads to low calcium levels. While hypoparathyroidism primarily affects calcium, it can also indirectly affect sodium and potassium levels through complex hormonal interactions and renal effects.

Pseudohypoparathyroidism: A rare genetic disorder characterized by resistance to parathyroid hormone, leading to hypocalcemia. It can also present with other electrolyte abnormalities due to the complex interplay of hormonal resistances.
Rhabdomyolysis: Although primarily known for causing hyperkalemia and hypocalcemia due to muscle breakdown, rhabdomyolysis can also lead to hyponatremia in the acute phase due to fluid shifts and renal impairment. It's a critical condition that requires immediate attention.

Barter Syndrome and Gitelman Syndrome: These are rare genetic disorders affecting the kidneys' ability to reabsorb electrolytes. They can lead to hypokalemia (not hyperkalemia, making them less likely in this scenario) and hypocalcemia, along with other electrolyte imbalances. However, they are worth considering in the differential diagnosis of electrolyte disturbances, especially in younger patients or those with a family history of similar conditions.
Liddle Syndrome: Another rare genetic disorder that affects the kidneys' handling of sodium and potassium, leading to hypertension, hypokalemia, and sometimes metabolic alkalosis. While it doesn't directly cause low sodium, high potassium, and low calcium, its inclusion in the differential for electrolyte imbalances is important due to its potential for severe hypertension and renal complications.