Hand, Foot, and Mouth Disease (HFMD): This is the most likely diagnosis due to the characteristic presentation of rash on the hands, feet, and mouth, which can sometimes extend to the elbows. HFMD is a common viral illness caused by coxsackievirus A or enterovirus 71.

Erythema Multiforme: This condition can present with similar skin lesions and can be triggered by viral infections or other factors. It's less common than HFMD but should be considered, especially if the rash has a distinctive target-like appearance.
Varicella (Chickenpox): Although the primary presentation of chickenpox is more widespread, it can start with localized rashes and could potentially be confused with HFMD, especially in the early stages.
Allergic Contact Dermatitis: This could cause a rash on the hands and potentially other areas if there's exposure to an allergen, but it's less likely to have the specific pattern of HFMD.

Meningococcemia: This is a severe bacterial infection that can cause a rash, often starting on the extremities. It's critical to identify quickly due to its high mortality rate if not treated promptly.
Stevens-Johnson Syndrome/Toxic Epidermal Necrolysis: These are severe skin conditions usually triggered by medications or infections, characterized by widespread skin necrosis and detachment. Early recognition is crucial for management and preventing severe outcomes.

Kawasaki Disease: This is an acute febrile illness of childhood that can cause rash, among other symptoms like fever, conjunctivitis, and lymphadenopathy. It's rare but important to consider due to its potential for cardiac complications.
Enteroviral Exanthems (other than HFMD): There are various enteroviruses that can cause rashes, and while HFMD is the most common, other enteroviral infections could present similarly, albeit rarely.