Meningococcemia: This condition is characterized by the sudden onset of fever, altered mental status, and a petechial rash, which can progress to purpura. The patient's hypotension that is unresponsive to fluid resuscitation, along with hyponatremia and hyperkalemia, further supports this diagnosis. Meningococcemia can lead to severe sepsis and septic shock, which would explain the hypotension and abdominal pain due to potential adrenal hemorrhage (Waterhouse-Friderichsen syndrome).

Severe Sepsis/Sepsis Syndrome: The patient's presentation with fever, altered mental status, hypotension unresponsive to fluids, and generalized abdominal pain could be indicative of severe sepsis or sepsis syndrome due to a bacterial infection other than Neisseria meningitidis.
Disseminated Intravascular Coagulation (DIC): The presence of a petechial rash, along with the patient's critical condition, could suggest DIC, which can be triggered by severe infections, including meningococcemia, and can lead to hyponatremia and hyperkalemia due to renal involvement.
Adrenal Insufficiency: The combination of hyponatremia, hyperkalemia, and hypotension that is unresponsive to fluid resuscitation could suggest adrenal insufficiency, potentially due to Waterhouse-Friderichsen syndrome in the context of meningococcemia or another severe infection.

Toxic Shock Syndrome: Although less common, toxic shock syndrome (caused by Staphylococcus aureus or Streptococcus pyogenes) can present similarly with fever, rash, hypotension, and multi-organ involvement. It is crucial to consider this diagnosis due to its high mortality rate if not promptly treated.
Rickettsial Infections: Infections like Rocky Mountain spotted fever can present with fever, rash, and severe illness, including altered mental status and hypotension. These are less common but critical to diagnose due to their severity and response to specific treatments.

Thrombotic Thrombocytopenic Purpura (TTP): Although rare, TTP could present with altered mental status, fever, and a petechial rash. However, the primary distinguishing feature of TTP—thrombocytopenia—is not mentioned in the scenario.
Vasculitis: Certain forms of vasculitis, such as Henoch-Schönlein purpura or ANCA-associated vasculitis, could present with rash, abdominal pain, and renal involvement leading to electrolyte imbalances. However, these conditions are less likely given the acute onset and severity of the presentation.
Hemophagocytic Lymphohistiocytosis (HLH): This rare condition involves an overactive and inappropriate immune response, which can lead to fever, rash, and multi-organ failure. It is a diagnosis of exclusion and would require specific laboratory findings to support it.