The baby, Jarrah, who was born in a refugee camp in Syria less than three weeks ago, had a lung infection and died at about 1.30pm on Thursday, having been taken to hospital.
Begum, who left east London to join ISIS in Syria when she was 15, made worldwide headlines last month as she publicly pleaded with the United Kingdom government to be allowed to return.
She wanted to return to Britain but was stripped of her citizenship.
A British government spokesman said: "The death of any child is tragic and deeply distressing for the family". However, the Bangladesh foreign ministry said in a statement that Begum is not a Bangladeshi citizen, nor has she ever visited the country.
"I don't really want to stay here, I don't want to take care of my child in this camp because I'm afraid he might even die in this camp", she said in an interview with United Kingdom broadcaster Sky News shortly after giving birth.
Begum's son is one of scores of children who have died fleeing from fighting in ISIS' last remaining Syrian enclave Baghouz.
Last month, Mr Javid confirmed the boy was a British citizen and said he had considered the child's interest when deciding to revoke Ms Begum's citizenship.More news: Kate Beckinsale's Ex Sends Warning to Pete Davidson
The International Rescue Committee on Friday said that 12,000 women and children had arrived at the Kurdish-run Al-Hol camp since Wednesday. The family has said it plans to challenge Javid's decision.
Begum was 15 when she travelled to Syria with two friends to marry IS fighters in 2015.
Begum's family had announced the boy's birth on February 17, saying they believed he was "in good health".
He told HuffPost UK that Begum had been moved to a new area of Syria and was living in a refugee camp: "We know she was living in a tent with a newborn baby in a camp with thousands of other people with very limited medical and food supplies".
"I have nothing but sympathy for the children that have been dragged into this".
Begum's lawyer Mr Akunjee said: "I confirm that it is our understanding that Shamima has been moved from Al-Hol due to safety concerns around her and her baby".
'Inside Syria, whether in a camp or maybe somewhere else, there is no British consular presence'.