The 73-year-old father-of-three, who had represented the constituency since 2010, is understood to have died from natural causes on January 7.
Ms Hamilton, who worked for the NHS for 25 years and re-joined the healthcare frontline during the pandemic to support the vaccine rollout, has been a local councillor for more than 17 years.
She said Mr Dromey was a “wonderful man” and the by-election had come about “in the saddest of circumstances”.
“First and foremost, I want to pay tribute to Jack Dromey,” Ms Hamilton said.
“Jack was a wonderful man and he’ll be sorely missed in Erdington. He was passionate about Erdington and cared deeply about everyone who lives here.
“On a day like today I’m thinking most of all about Jack’s family and friends. We are all still grieving and this by-election has come about in the saddest of circumstances.”
The Erdington resident said she understands the issues faced by “every local family”, adding she would “fight for every vote”.
“I’ve got a track record of getting things done. My priority will be to restore community pride in Erdington – by tackling fly-tipping and regenerating the high street.
"I will address the issues around HMOs (house in multiple occupation) and high levels of anti-social behaviour,” she said.
“I live here so this is deeply personal for me. I will fight for every vote and I’m looking forward to speaking to as many people as possible over the coming weeks.”
MPs from across the political spectrum paid tribute following Mr Dromey’s death and sent their condolences to his wife, the party grandee Harriet Harman.
Born in London in 1948, Mr Dromey had three children with Ms Harman, who announced last month she will stand down as an MP after the next general election.
At the 2019 general election, Mr Dromey defended his seat by 3,601 over the Conservatives in second place.
The seat has been held by Labour since its establishment in 1974. The date of the upcoming by-election is not yet known.