Former Delaware Attorney-General Beau Biden died on May 30 from brain cancer at the age of 46.
Beau Biden was seen as a rising star of US politics but suffered from health problems in recent years.
He had intended to run for Delaware state governor in 2016.
Barack Obama said: “Beau Biden was an original. He was a good man. A man of character. A man who loved deeply and was loved in return.”
About 1,000 people – including Bill and Hillary Clinton and numerous other top politicians – attended the funeral at a Roman Catholic church in Wilmington, the largest city in the state of Delaware.
Joe Biden led a procession into the church with his family at the beginning of the service.
Mourners heard Barack Obama describing Beau Biden as a public servant who learned through early tragedy what mattered most and as a result decided upon living “a life of meaning” that would inspire those around him.
“He was a scion of an incredible family, who brushed away the possibility of privilege for the harder, better reward of earning his own way,” the president said.
Barack Obama described Beau Biden as a son, a father, a soldier and a politician who refused to take short cuts in his determination to serve his country and others.
He said that a “cruel twist of fate” killed Beau Biden’s mother and infant sister in a car crash four decades ago and left Beau – three years old at the time – and his younger brother Hunter in hospital.
Coldplay singer Chris Martin was a soloist at the service – he volunteered to perform after hearing that Beau Biden liked the band.
Beau Biden was diagnosed with brain cancer in August 2013 and underwent treatment that was initially successful. However, the cancer recurred earlier this year.
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