The Foyle MLA said whoever succeeded Mr. Brokenshire needed to appoint a neutral chair to take charge of forthcoming talks geared towards restoring power-sharing.
She joined the Conservatives and in 2010 won the seat of Staffordshire Moorlands from Labour, becoming a government whip three years later and a junior minister in 2014.
"While it is not always obvious to the public gaze, very important progress has been made on significant issues over the previous year and I believe that a positive outcome can still be achieved".
The Northern Ireland Office confirmed that Mr Brokenshire has resigned, but would not confirm reports that his decision was based on health grounds.
In a letter to Brokenshire, May said she was sorry to hear about the medical complication and commended the secretary on "what a good job" he would have done.
Trump to attend WEF gathering in Davos
Last year's attendees included Google co-founder Sergey Brin, British Prime Minister Theresa May and Chinese President Xi Jinping. Trump waves as he walks across the South Lawn of the White House to depart by Marine One, in Washington, DC, USA, 16 June 2017.
"In the course of negotiations, the SDLP has often disagreed with the position of his government but I have always found Mr Brokenshire to be personable, reasonable and, so far as possible, helpful".
Brokenshire, who has led government efforts to restore the devolved Northern Irish executive after power sharing collapsed past year, was stepping down ahead of major surgery in coming weeks, his spokesman said.
"I have a small lesion in my right lung which needs to be removed". When she was named culture secretary after Ms May became prime minister in 2016, the cultural and sports worlds initially rejoiced.
She wished him well with his surgery and said he "leaves the role with a very intimate knowledge of Northern Ireland and I look forward to working with him again in the future". He was not among ministers who were predicted to go in the reshuffle.
It is likely to represent her biggest overhaul of her top team since she appointed her first Cabinet on entering No 10 in 2016. He formerly served as immigration minister under former Prime Minister David Cameron.