The 19-year-old, from Staffordshire, died on Wednesday after a four-year battle with the disease.
Stephen's appeal in aid of the Teenage Cancer Trust on his Just Giving page has now raised £3.7m.
There are now plans to make a film about his life, using footage shot over the past 12 months.
Film-maker Grigorij Richters, 26, from Hamburg, Germany, met Stephen at the Champions League final at Wembley in May 2013 and they became friends.
The visit to Wembley was on the "bucket list" of 46 ambitions Stephen wanted to achieve before he died, along with raising money for the Teenage Cancer Trust.
The pair agreed to film the bucket list together and he now has hours of footage, which he wants to use to tell Stephen's story.
Mr Richters said Stephen was "a young man who did the inconceivable, the unbelievable, at a young age".
At his former school, Chase Terrace Technology College in Burntwood, people have queued to sign a book of condolence and leave flowers.
Tributes have also been written in chalk on pavements in Birmingham.
Suggestions on social media that there should be a permanent memorial in the city will go before the Broad Street Walk of Stars committee in the summer.
Mike Olley, the manager of Broad Street in Birmingham, where the stars are located said: "We are always happy to take suggestions for the Walk of Stars.
"Many people have nominated Stephen now and I am delighted to put this before the committee."
An appeal started by the cancer fundraiser Stephen Sutton has raised a further £400,000 - only a day after the teenager died.