The change indicates that another attack may be imminent.
May said the U.K. could not ignore the possibility that more individuals are linked to the Manchester Arena attack.
A photo of Abedi was first published on the front page of the British newspaper The Sun.
The suspect was originally identified as a 23-year-old male, according to a source; police have since said he was 22.
A 23-year-old man has been arrested in South Manchester in connection with the attack, police said. Police also said authorities executed two warrants as part of the investigation: one in the Whalley Range district of Manchester and one in the Manchester suburb of Fallowfield, where a controlled explosion took place.
Greater Manchester Police are requesting dashcam footage from "anyone who was in Manchester city centre" between 8 p.m. and 11 p.m. Monday as part of its investigation.
A 39-year-old man was also arrested in Birmingham today near a vigil for the attack. The man is "known to police and is thought to have a history of mental ill-health," Birmingham police said in a statement. A small ax and large stick were recovered at the vigil, police said.
ISIS has claimed responsibility for the explosion. In a statement translated from Arabic, ISIS said that a "soldier of the caliphate" placed explosives at a gathering of "crusaders" — meaning Christians — at the Manchester Arena. The statement said about 30 people were killed and about 70 more were wounded.
Greater Manchester Police said officers were called to the Manchester Arena just before 10:35 p.m. local time on Monday. The explosion happened near the arena's foyer after the concert, according to witnesses, who reported hearing a bang as they exited.
An 8-year-old girl named Saffie Rose Roussos was among those killed.
Twelve other children under the age of 16 were seriously injured, officials said.
"We struggle to comprehend the warped and twisted mind that sees a room packed with young children not as a scene to cherish but as an opportunity for carnage," May said today.
"I had my best friend with me, and I grabbed hold of her wrist and told her never let go of me," he said. "We just ran. We jumped over chairs, railings to get out of the doors. We had to force open doors that wouldn't open because people were trying to get to — the entire capacity of the 20,000-person arena were trying to get out of one exit."
"It felt like an eternity," Harries said, but it "couldn't have been more than two, three minutes from in our seats to outside of the arena."
The Greater Manchester Police tweeted today that officials do not believe there are any unaccompanied children at Manchester-area hotels.
"I want to thank all the members of the emergency services, who have responded with such professionalism and care," she continued. "And I would like to express my admiration for the way the people of Manchester have responded, with humanity and compassion, to this act of barbarity."
A spokesperson for the Manchester Arena tweeted today, "Last night, our community suffered a senseless tragedy. Our entire team's thoughts and focus are now on supporting the people affected and their families.
"We are assisting the police in any way we can. We cannot praise the emergency services enough for their response and have been inspired by the way the people of this great city of Manchester rallied round last night and have continued to respond today. It shows the very best of this city."
ABC News' Mike Levine and Lena Masri contributed to this report.