Police arrested Ezzedine Khaled at his apartment on Saturday and he was remanded into custody for four days, his lawyer Mahmoud Ottman told AFP.
Khaled is a member of a group called Street Children that posts satirical songs on its Facebook page.
Its latest appears to have touched a nerve as police round up activists involved in April protests against President Abdel Fattah al-Sisi for handing over two islands to Saudi Arabia.
Many Egyptians believe the islands in the Straits of Tiran belong to them, although the government says they were Saudi and had been leased to Egypt in the 1950s.
The issue galvanised dissidents who say Sisi, a former military chief who overthrew his Islamist predecessor Mohamed Morsi in 2013, now rules the country with an iron fist.
In the video, Khaled and other group members, who appear to be in their 20s, sarcastically interrogate one another.
“What do you have to say about the charges against you?” one of them asks.
“I’m not (Muslim) Brotherhood! I’m a womaniser,” another responds, referring to Morsi’s Islamist movement.
They then break out into a song about the devaluation of the Egyptian pound and the island deal.
In another clip they mock Sisi and the police, singing: “Leave, have some shame.”
Ottman said his client is accused of “inciting protests and publishing a video that insulted state institutions.”
A prosecution official confirmed Khaled’s detention.
The overthrow of Morsi, who was deeply unpopular, unleashed a police crackdown on his supporters that has killed hundreds of protesters and imprisoned thousands of people.
The crackdown has extended to liberal and leftwing dissidents.
Morsi’s ouster also set off a jihadist insurgency that has killed hundreds of soldiers and policemen, including eight who were gunned down south of Cairo overnight.