The 19-year-old will play Freddie Slater, son of Maureen “Little Mo” Slater, and is currently filming scenes scheduled to air this summer.
The character of Freddie was last seen in Walford in May 2006 when he and his mother left to start a new life.
Now he is returning to Albert Square, aged 18, and finds himself on a path that could lead to heartbreak.
Big Brother star Goody, who died in 2009 aged 27 after being diagnosed with cervical cancer, had Bobby and his brother Freddy, now 17, with fellow reality TV star Jeff Brazier.
She was an avid fan of EastEnders and in 2006 reportedly petitioned two of its stars to talk to their bosses about securing her a role during a chance meeting at V Festival.
Bobby Brazier said: “I’m really excited to be joining the cast of EastEnders - so far it’s been perfect. Everybody has been so beautiful to me and made settling in easier than I could’ve hoped.
“Stepping into the acting world has always been a vision of mine and doing that with EastEnders as a Slater is a blessing, I’m very grateful.
“Loving every minute of Freddie so far and can’t wait to see what’s in store for him in Walford.”
Describing his character, show bosses said: “A bundle of energy and a truckload of trouble, Freddie Slater has a heart of gold but can’t help getting himself into mischief.
“While he’s sweet like his mum, he certainly isn’t shy and manages to cause a stir wherever he goes - but is he a true Slater at heart?”
Chris Clenshaw, EastEnders’ executive producer, said: “Freddie Slater is a chaotic 18-year-old with a sensitive soul.
“A modern-day lad who’s in touch with his spiritual side. He’s philosophical, accepting, but full of mischief and makes mistakes.
“Bobby is an amazing young addition to the EastEnders cast who brings heart and humility to the role of Freddie Slater and I can’t wait for audiences to get to know him.”