The young child was crushed in his sleep early Monday when a mud wall collapsed onto his family's shanty in Kutupalong refugee settlement, local police chief Abdul Khaer told AFP.
"It was triggered by rain over the last three days," Khaer said.
Kutupalong camp magistrate Rezaul Karim said about 300 shelters had been damaged and dozens of refugees evacuated since the first downpours of the rainy season began in earnest Saturday.
"We relocated 20 families last night," he told AFP.
Aid agencies have been warning that the monsoon could unleash "an emergency within an emergency" for the close to one million Rohingya living in temporary shelters in southeast Bangladesh.
The region is prone to cyclones, landslides and devastating downpours during the rainy season, which arrives in June and typically lasts three months.
The district where refugees perch on steep hillsides in bamboo and plastic tents was hammered by 138 millimetres (5.4 inches) of rain between Saturday night and midday Sunday, Bangladesh's meteorological office said.
The area is forecast to receive 2.5 metres of rainfall during this year's monsoon -- roughly triple what Britain gets in a year.
"Some areas, like the football field areas, are flooded. Some houses have been inundated with water. There have been a few landslides. The conditions are bad," UN refugee agency spokeswoman Caroline Gluck told AFP on Sunday.
The monsoon season "is going to be big test for everybody involved in the humanitarian response", she added.
Last year, monsoon rains triggered landslides in Cox's Bazar and the nearby Chittagong Hill Tracts, killing at least 170 people. More than 100 died in landslides in the region in 2012, and two years earlier around 50 perished.
An estimated 200,000 refugees are at direct risk of landslides and floods and need relocating, aid agencies working in the camps say.
Nearly 29,000 have been moved to new locations but "more evacuation space is urgently needed to save lives", Kevin J. Allen, head of UNHCR's operations in Cox's Bazar, posted on Twitter on Sunday.
There are fears flooding could cause latrines to overflow and spread disease in the camps where around 700,000 Rohingya have arrived since August, fleeing an army crackdown.
They joined hundreds of thousands of earlier refugees from mainly Buddhist Myanmar, where the Rohingya are a persecuted and stateless minority.
Many of the hillsides were cleared of trees to make way for shelters, making the land highly unstable.
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