Matthew Young developed the ailment four days after his birth at Warrington General Hospital on 25 April this year.
His GP prescribed peppermint water, which was supplied by Boots chemist at the Hallwood Health Centre.
Matthew's father, Neil Young, said the child turned blue after he used a syringe to administer the water by drops at the family home in Runcorn, Cheshire.
Mr Young told the Liverpool inquest that he believed the baby was choking and tried to clear his airways. He gave him mouth-to-mouth resuscitation while the child's mother telephoned for an ambulance.
"I tipped Matthew over and tapped his back and he had saliva around his mouth. I gave him mouth-to-mouth and felt a burning sensation on the back of my tongue. I had a blister on the back of my tongue," Mr Young said.
The baby was taken to Alder Hey Children's Hospital in Liverpool, where doctors diagnosed Matthew as having severe brain damage and he died 18 days later.
The Liverpool coroner, Roy Barter, adjourned the inquest until a date to be arranged.
A spokeswoman for the Cheshire Constabulary said yesterday: "On 29 April, Matthew Young, born on 25 April, was admitted to Alder Hey hospital. Sadly he died on 17 May.
"Cheshire police are fully aware of the circumstances in which he died and a full investigation will take place."
A spokesman for Boots chemist said yesterday: "Our deepest sympathy goes out to Matthew's family following this tragic news.
"In the light of the coroner's inquest it would be inappropriate for us to comment any further at this stage."