Urinary Tract Infection (UTI) or Pyelonephritis: Given the abdominal pain and singular kidney, a UTI or pyelonephritis is a common and likely diagnosis. The elevated eosinophils could be a secondary response to the infection.

Nephrolithiasis (Kidney Stones): Although less common in children, kidney stones can cause severe abdominal pain and could potentially lead to an elevation in eosinophils due to an allergic or inflammatory response.
Acute Interstitial Nephritis: This condition, often caused by an allergic reaction to medications or infections, can lead to abdominal pain and elevated eosinophils.
Parasitic Infections: Certain parasitic infections, such as those caused by worms, can lead to elevated eosinophils and abdominal pain, though this might be less common in areas with good sanitation.

Appendicitis: Although the presence of a singular kidney might shift the clinical suspicion towards renal causes, appendicitis is a surgical emergency that cannot be missed. The location of the pain and other symptoms like fever and vomiting should be carefully evaluated.
Intestinal Obstruction: This is another surgical emergency that could present with abdominal pain and potentially elevated eosinophils if there's an associated infection or allergic response.
Vasculitis (e.g., Henoch-Schönlein Purpura): Vasculitis can cause abdominal pain, elevated eosinophils, and renal involvement, making it a critical diagnosis not to miss due to its potential for serious complications.

Lymphoma: Although rare, lymphoma can cause abdominal pain and elevated eosinophils. Given the age of the patient and the singular kidney, this would be an unusual diagnosis but should be considered if other tests are inconclusive.
Eosinophilic Gastroenteritis: A rare condition characterized by eosinophilic infiltration of the gastrointestinal tract, which could explain the abdominal pain and elevated eosinophils.

Complete Blood Count (CBC): To confirm the elevation in eosinophils and assess for other abnormalities.
Blood Chemistry: Including renal function tests to assess kidney function.
Urinalysis: To check for signs of infection or kidney stones.
Imaging Studies: Ultrasound or CT scan of the abdomen to evaluate the singular kidney and potential causes of abdominal pain.
Stool Tests: To rule out parasitic infections if suspected.
Allergy Testing: If an allergic cause of elevated eosinophils is suspected.
Biopsy: In cases where interstitial nephritis or other specific conditions are suspected and cannot be diagnosed through less invasive means.