Humam Balawi is said to have been a Jordanian doctor turned double agent, having originally been recruited by the Jordanians to spy on al-Qaeda.
His father, Khalil Balawi, said that the man who appeared on the video was definitely his son.
Speaking in Jordan, he blamed "American oppression against the Muslim world" for his son's act.
'Blood of our leader'
In the video clip shown on al-Jazeera, Humam Balawi says he is sending a message to the "enemies of Islam", who he names as Jordan's intelligence services and the CIA.
Humam Balawi blew himself up at a CIA base in Khost province, Afghanistan
"We say that we will never forget the blood of our leader, Baitullah Mehsud," he says.
"What remains is to avenge him inside and outside America."
The Jordanian, speaking in Arabic, says the Pakistani Taliban leader paid with his life for offering to shelter al-Qaeda leader Osama bin Laden.
He appears to dismiss the idea he himself was in the service of the Jordanians or US, saying that a Muslim jihadist "will not sell his religion".
The IntelCenter, a US-based group monitoring extremist sites, said the video had been released by the Pakistani branch of the Taliban.
The bombing of the American base was an audacious attack which sent shockwaves through Western intelligence services, the BBC's Orla Guerin in Islamabad reports.
So far there has been no comment from the authorities in Pakistan, she says, but the video is bound to increase pressure on them to crush militant groups on their soil who attack Western forces across the border in Afghanistan.
Baitullah Mehsud was killed in a US missile attack in the tribal region of South Waziristan, which borders Afghanistan, in August last year.
Two months later, the Pakistani military launched a major offensive in the region.
More than 600 people have died in militant attacks in Pakistan since then, most of them believed to be in reprisal for the army's campaign against the Taliban.
Correspondents say Hakimullah Mehsud is thought to be in South Waziristan.
Forward Operating Base Chapman, a former Soviet military base in Afghanistan, is used not only by the CIA but also by provincial reconstruction teams, which include both soldiers and civilians.
The airfield is reportedly used for US drone attacks on suspected militants in neighbouring Pakistan.
FOB Chapman operates from Khost Airfield 32km from Pakistan border
Former Soviet base is reportedly used for launching US drones
Airfield extended to allow C-130 transporter planes to land
Named after Nathan Chapman, first US soldier killed in Afghanistan in 2002
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