They evolved into a wider movement for democratic reform and a halt to eroding freedoms, with protesters' anger fuelled by Lam's refusal to publicly acknowledge their grievances.
The semi-autonomous southern Chinese city has been wracked by two months of protests and clashes between police and increasingly hardcore demonstrators in the biggest threat to Beijing's rule since its 1997 handover from Britain.
In a press conference, Lam on Monday warned the city was nearing a "very dangerous situation" as she framed the protests as a challenge to China's sovereignty.
Under the terms of the 1997 handover deal with Britain, Hong Kong has rights and liberties unseen on the Chinese mainland, including an independent judiciary and freedom of speech.
But many say those rights are being curtailed.
Protesters are demanding the resignation of city leader Carrie Lam, an independent inquiry into police tactics, an amnesty for those arrested, a permanent withdrawal of a reviled extradition bill, and the right to elect their leaders.
The remark has come a day after a series of rallies that included bloody clashes between protesters and pro-government men brandishing poles