Ken Saro-Wiwa Jr, 47, passed away after suffering a stroke, his family say.
He was a journalist who became an adviser to three presidents.
The 1995 execution of his father by a military government for leading protests against environmental degradation caused by the oil industry sparked global outrage.
Saro-Wiwa Sr led the Movement for the Survival of the Ogoni People (Mosop), which accused oil multinational Shell of destroying the environment in his home region of Ogoniland in south-eastern Nigeria.
His execution after a secret trial under Gen Sani Abacha led to Nigeria being suspended from the Commonwealth.
Noo Saro-Wiwa, sister of the late journalist, told the BBC: "It is with great sadness that we announce that Ken Saro-Wiwa Jr passed away suddenly. His family are devastated and request privacy at this difficult time."
Funeral arrangements are yet to be worked out, the family says.
Ken Saro-Wiwa was first appointed in 2006 as a special adviser on peace and conflict resolution by former Nigerian President Olusegun Obasanjo.
He later served Mr Obasanjo's successor, President Umaru Yar'Adua, as an adviser on international affairs and stayed on under President Goodluck Jonathan until he lost last year's election.
His willingness to work with the federal government marked him out as less militant than his father.
But like his father, he was committed to the cause of the Ogoni people.
In a 2015 opinion piece for the UK's Guardian newspaper, he wrote that the effects of the oil pollution on Ogoniland had still not been cleared up.
"If my father were alive today he would be dismayed that Ogoniland still looks like the devastated region that spurred him to action.
"There is little evidence to show that it sits on one of the world's richest deposits of oil and gas."
A 2011 UN report said Nigeria's Ogoniland region could take 30 years to recover fully from the damage caused by years of oil spills. The study said complete restoration could entail the world's "most wide-ranging and long-term oil clean-up".
It added that communities faced a severe health risk, with some families drinking water with high levels of carcinogens.
Shell has accepted liability for two spills and said all oil spills were bad for Nigeria and the company.