Harry Sievey was one of two cyclists involved in a collision with a Vauxhall Corsa in Mauldeth Road, Withington, on Sunday.
Hundreds of online tributes have been paid to the 24-year-old musician.
A 61-year-old man has been arrested on suspicion of causing death by dangerous driving, Greater Manchester Police said.
Mr Sievey was well known on Manchester's music scene. His late father found fame in the 1980s with his distinctive comedy character and his large papier mache head.
His family said in a statement: "There are no words for another tragic and sad loss to our family.
"Our little brother and loving son Harry was always happy-go-lucky. We are heart broken."
The crash happened in Mauldeth Road near the junction of Wellington Road shortly before 22.15 GMT.
The other cyclist, in his 20s, was taken to hospital with minor injuries.
Neil Taylor, who helped bring a statue of Frank Sidebottom to his hometown of Timperley, said the news about Harry had "come as a complete shock".
"Harry was so young and talented, and reading the tributes to him show what a popular young man he was," he said.
Writer Jon Ronson, who wrote a film about Mr Sievey's father and was also in his band, tweeted: "Terrible news that the delightful Harry Sievey, son of Chris, died last night in a bicycle accident. RIP."
Altrincham FC also expressed its "heartfelt condolences".
Manchester's arts centre Home said Mr Sievey was "a talented musician and member of our bar staff team".
Sgt Lee Westhead, of Greater Manchester Police, said:  "It is key we now try and establish exactly what happened and I would urge anyone who saw the collision to please call my team."
Chris Sievey died of cancer in 2010 aged 54. A statue of his comic creation was later unveiled in Timperley.
Mr Sievey, under the guise of Sidebottom, made regular appearances on television throughout the 1980s, presented his own show and also featured on radio.
The character, who also released a number of cassettes and 7-inch singles, was said to have inspired the 2014 film Frank, starring Michael Fassbender.
The son of comedy character Frank Sidebottom's creator Chris Sievey has been killed in a crash in Manchester.