The arrest came after the prosecutor general, who is investigating charges of inciting violence, issued a warrant for al-Shater, al-Masry al-Youm reported.
Khairat al-Shater was arrested after the ousting of President Mohamed Morsi in early July.
He allegedly incited violence between supporters and opponents of Morsi after the overthrow.
Saad's brother, Hassan al-Shater, tweeted about the arrest and prayed for his father and brother, who are now both detained.
Saad's sister, Somaya, also spoke to the media.
"More than 30 people, most of whom were masked and were carrying machine guns broke [down] the door into the apartments of my father and my sister to search for 22-year-old Saad, asking about him by name," she said.
The family claims that the security forces asked for Saad's identification and then beat him.