Bilawal says his father will run the party while he is away
He will become president in a ceremonial capacity while he finishes his studies at Oxford University.
Bilawal told journalists at the Bhutto family home: "My mother always said democracy is the best revenge".
Ms Bhutto's widower, Asif Ali Zardari, who will run the party day-to-day, said it would contest January elections.
And he appealed to the former prime minister Nawaz Sharif - a long-time Bhutto rival - to drop his threat to boycott the polls.
Mr Zardari and his son were speaking at a news conference after a meeting of the PPP leadership in Naudero, near Larkana in southern Pakistan.
Another senior party official, vice-chairman Makhdoom Amin Fahim, said Ms Bhutto had named Mr Zardari as her successor as party chairman.
But he said Mr Zardari had turned it down in favour of his son - a decision he said the party leadership had endorsed.
Mr Zardari also announced that the couple's children would now change their names and be called Bhutto Zardari.
Sitting between his father and Mr Fahim, Bilawal himself said his father would run the party while he was away at university.
"When I return, I promise to lead the party as my mother wanted me to," he said.
But Mr Zardari blocked any further reporters' questions to Bilawal, saying that although party chairman, he was still of "tender age".
"We are all in mourning," he said.
Mr Zardari also said he had refused to allow an autopsy on Ms Bhutto's body.
"I've lived here long enough to know how and where an autopsy would have been conducted," he said.
Instead, he said the party was asking the United Nations and the British government to conduct an investigation similar to the one carried out after the killing of the former Lebanese Prime Minister Rafik Hariri.
And he appealed for an end to the unrest in Pakistan, which has killed at least 38 people since Ms Bhutto's killing.
Earlier, Pakistani television released new pictures it said showed Ms Bhutto's attackers - a gunman and a suicide bomber. They also apparently showed Ms Bhutto was inside her car, and no longer standing through the sun roof, when the explosion happened.
The images added to the dispute over Ms Bhutto's death.
Interior ministry spokesman Brig Javed Iqbal Cheema said on Friday that she was killed when the force of the bomb blast knocked her head against a sun roof fitting, and was not hit by bullets.
The PPP has insisted she was killed by two bullets, one of which pierced her skull and another which hit her in the neck.
The assassination opened the question of whether elections due on 8 January would go ahead as planned.
The ruling party says they are likely to be delayed for several weeks, on the grounds that the vote would "lose credibility" if held under current conditions.
Opposition parties have been calling for a delay, amid widespread unrest and political disarray following the murder of Ms Bhutto.
Tariq Azim of the ruling PML-Q party said a delay would allow the PPP more time to re-organise.
Pakistan's election commission has called an emergency meeting for Monday, to decide whether the poll should be delayed.
But the PPP says it wants the elections to go ahead as planned - even though it is not clear who would be its leading candidates.
At 19, Bilawal is legally too young to stand for parliament.
And his father has been repeatedly accused of corruption - though he denies the charges and has never been convicted in court.
Mr Zardari said party vice-chairman Mr Fahim would probably be its candidate for prime minister.But the BBC's Owen Bennett-Jones - reporting from Naudero - says filling the political gap left by Benazir Bhutto will be a very big challenge for her party.