Three explosions set off by suicide bombers rocked the cemetery where Salim Ezadyar, the son of Senator Alam Ezadyar, was about to be buried, said Najib Danish, Interior Ministry deputy spokesman.
Ezadyar was one of several people killed Friday when police clashed with protestors who were demanding better security in the capital.
Mr Ezadyar was the son of Alam Ezadyar, the deputy speaker of the upper house of Afghanistan's Parliament.
The commander for Kabul's garrison, General Gul Nabi Ahmadzai, in a news conference later Friday said some protesters carried weapons and wounded some security personnel.
At least four people died in Friday's protests, as police fired live rounds to disperse hundreds of demonstrators.
The type of the explosions has not been ascertained so far and no group has claimed responsibility behind the attack so far.
On Friday, demonstrators rushed toward police who fired warning shots and when they attempted to move closer to the palace, police sprayed them with hoses from a water tanker and later fired tear gas.
All roads toward the palace and nearby diplomatic areas have been blocked by police and there was limited movement of vehicles and people.
Most of the casualties from the truck bombing were civilians, including women and children, officials have said.
"We reiterate our solidarity with friendly and brotherly people and Government of Afghanistan in the face of these terror attacks that aim to disturb peace and public order in the country", Turkish Foreign Ministry said on Saturday.
Afghan President Ashraf Ghani chaired a meeting of high-ranking security officials earlier on Saturday to discuss the deteriorating security situation in the country. He also called on members of the public not to join the demonstration.
Afghan policemen arrive at the site of blasts in Kabul, Afghanistan June 3, 2017.
Isis has been attempting to expand its presence in Afghanistan, claiming a separate bombing on Wednesday in Jalalabad that killed a Taliban member, and has called for increased global terror attacks during Ramadan.