The 19-year-old super-lightweight will face an as-yet unnamed opponent before Joshua fights American Charles Martin.
Benn Jr's goal is to surpass his father, who held the WBO middleweight and WBC super-middleweight titles.
"Until I've become the youngest world champion or won more belts, I'll always be Nigel Benn's son," he said.
"Achieving more will be very hard."
Benn Jr has signed a deal with Matchroom Boxing, and his father is confident he can succeed - having "thrown the kitchen sink at him" in a sparring session.
The teenager said: "My dad didn't really want me to fight. He thought that I would give it a try and have it as a hobby, but it escalated really quickly.
"He said there were a million other things to do. But I have no choice but to fight because it's in me. It's what I want to do. I want to be the best."
Nigel Benn won 42 of his 48 bouts and twice fought Chris Eubank, whose son - also called Chris - won the British middleweight title on Saturday.
Nigel Benn's son Conor will make his professional debut on the undercard of Anthony Joshua's world title fight at the O2 Arena on 9 April.