A Texas congressman named the child as Felipe Alonzo-Gomez.
It is the second time this month an immigrant child has died while being detained after crossing the US-Mexico border.
The previous instance saw seven-year-old Jakelin Caal, also from Guatemala, die just hours after being taken into custody.
Texan congressman Joaquin Castro called for a congressional investigation into the boy's death.
"We must ensure that we treat migrants and asylum-seekers with human dignity and provide the necessary medical care to anyone in the custody of the United States government," he said.
"The administration's policy of turning people away from legal ports of entry, otherwise known as metering, is putting families and children in great danger."
Thousands of migrants have travelled from Central America to the US border.
The migrants say they are fleeing persecution, poverty and violence in their home countries of Guatemala, Honduras and El Salvador.
Many of them say their goal is to settle in the US despite warnings by US officials that anyone found entering the country illegally will face arrest, prosecution and deportation.
The boy died shortly after midnight on 25 December, US Customs and Border Protection said.
In a statement, the agency said the eight-year-old had displayed "signs of potential illness" on Monday.
It reported that he and his father were taken to a hospital in Alamogordo, New Mexico, where the boy was diagnosed with a cold and a fever, given prescriptions for amoxicillin and ibuprofen, and released on Monday afternoon.
He returned to the hospital on Monday evening after he began vomiting and died there just hours later, the statement added.
The agency said the cause of death had not been determined, and that the Department of Homeland Security's inspector-general and the Guatemalan government had been notified.
Jakelin was crossing the US-Mexico border with her father as part of the caravan of Central American migrants.
US officials said they had tried to cross the Mexico-US border illegally.
Jakelin and her father handed themselves in to US border agents in New Mexico in early December.
While in custody, Jakelin developed a high fever and died of liver failure several days later.
The Washington Post, who first reported her death, quoted border officials as saying she had died of liver failure brought about by dehydration and shock, and that she "reportedly had not eaten or consumed water for several days".
She was said to have had a body temperature of 105.7F (40.9C).
Jakelin's father Nery remains in the US, where he is awaiting a decision on whether he will be allowed to stay.
Her remains were returned to Guatemala this week, and her funeral took place in the village of San Antonio Secortez on Tuesday.
It's been running high since the arrival of thousands of migrants in recent weeks.
Last month, US border agents used tear gas on a crowd of migrants, including children, trying to cross the border.
The agents said that personnel had been assaulted and hit by stones.
However, critics accused the Trump administration of a draconian response, while Mexico demanded an investigation into the incident.
The migrants have travelled in large groups, dubbed "caravans", for more than 4,000km (2,500 miles) from Central America.
Among them are many families with young children.
President Donald Trump has vowed to keep each migrant on the Mexican side of the border until courts have decided their cases, meaning some face a long wait.
They have been spending time in temporary shelters in the Mexican border city of Tijuana and in Mexicali, 180km to the east.
