Robert Mugabe junior, who is thought to be 31, is accused of destroying cars and other property over the weekend.
He was arrested following a complaint from his friend, Sindiso Nkatazo, also 31, police said in a tweet.
Mr Mugabe junior, the second child of the former president with his widow, Grace, has not yet commented.
The police say the damaged property was valued at $12,000 (£10,000).
News site ZimLive posted a video of Mr Mugabe junior walking to Harare Magistrates’ Court wearing a suit and facemask, followed by a throng of people, as others stood on the sidelines taking pictures of him on their phones.
Former President Robert Mugabe died in 2019 at the age of 95, two years after being forced to resign following 37 years in power.
It is extremely unlikely that his son would have been arrested if his father was still president, analysts say.
Mr Mugabe was replaced by his former right-hand man Emmerson Mnangagwa.
Relations between him and the Mugabe family had been fraught after they accused him of betraying Zimbabwe’s long-time leader.
His widow, Grace, had harboured ambitions of succeeding her husband but was sidelined by supporters of Mr Mnangagwa.
However, Mr Mugabe junior has since reconciled with President Mnangagwa and last year attended a rally where he was publicly welcomed back into the ruling Zanu-PF party.