The youngster was seen by a GP and in A&E but not admitted, according to a report.
However when he was taken back to hospital, he was taken to intensive care and later died.
Hospital bosses are investigating the death, which has been detailed in a report about serious incidents at Leeds Teaching Hospitals NHS Trust in May and June.
Other incidents included a patient who was not properly monitored in the Emergency Department after taking an overdose and later died.
The report to hospital directors says there was a delay in diagnosing the child with meningitis.
“When the child re-presented to hospital he was admitted to the Paediatric Intensive Care Unit,” it says.
“Sadly, despite maximal support, the patient subsequently died.”
Investigators have been appointed to look into the case, while the report says the child’s family are being supported by hospital staff.
The name of the hospital has not been revealed, although care for children is usually provided at Leeds Children’s Hospital, on the Leeds General Infirmary site.
Three other patients died in separate incidents detailed in the report.
A patient who was being treated in a Leeds Emergency Department after taking an overdose did not receive cardiac monitoring and was found unresponsive after an hour. Despite resuscitation, they died six days later.
Another patient who had undergone a hip operation was due to have further surgery. They were admitted to hospital with signs of infection and treated for sepsis but later died.
A premature baby died after undergoing a blood transfusion. A verbal apology has been given to their family and the case has been reviewed.
Another baby was stillborn after the mother went to hospital several times over seven weeks for checks because the baby’s movements had reduced, according to the report.
Investigations are now taking place into all the incidents.
Two patients may also have had incorrect breast surgery after a problem processing samples taken through biopsies.
The biopsies were taken at Bradford Royal Infirmary, the specimens processed in Leeds and the surgery carried out at another hospital and an inquiry is underway to discover where the mistake happened.