Eight-year-old Robert Christie was airlifted to hospital in Belfast from the farm near Dunloy on Saturday, but doctors were unable to resuscitate him.
His 52-year-old father, also called Robert, was helping out on the farm on Ballynaloob Road, near his own.
Both were overcome by poisonous fumes as they were mixing slurry.
It is understood they were found by a postman on Saturday afternoon.
Dangerous gasesThe Health and Safety Executive is investigating.
A police spokesman said: "At the request of family members, police will not be releasing any further details at this time."
Emergency services were sent to the farm 
The accident happened on Saturday afternoon Barclay Bell, the deputy president of the Ulster Farmers Union, said dangerous gases build up in the slurry tanks.
"It's really a tank for holding all the waste products from animals produced during the winter time," he said.
"It stays in these tanks maybe for a number of months and a lethal combination of gases build up, including methane, carbon dioxide, ammonia, and worst of all, probably, hydrogen sulphide."
DUP MP Ian Paisley Jr said: "It is deeply distressing to learn of yet another death near a slurry pit."
He added: "These accidents are tragic as they remind us of the real dangers the farming community face every day."
Sinn Féin assembly member Daithi McKay said: "Local people here are shocked at what has happened earlier today, as are the farming community across north Antrim."
TUV assembly member Jim Allister said: "The dangers of farming are ever present and when they claim lives, then it comes home to us all just how vulnerable farming families can be."
It is the latest in a long line of fatal accidents involving slurry tanks on Northern Ireland's farms.
The most high profile incident was in September 2012, when Ulster rugby player Nevin Spence, his father Noel and brother Graham died after they were overcome by fumes on their family farm.