Ronan Sexton, 31, suffered a serious head injury after climbing on to the phone box outside Brighton Pier in East Sussex and then plunging to the ground.
Mr Sexton had been enjoying a night out with friends when the accident happened just after 4am on Sunday.
He was admitted to a local hospital but then transferred to Hurstwood Park neurological centre in Haywards Heath, West Sussex, in a critical condition.
Mr Sexton's mother Bernadette - the sister of Taken and Stars Wars actor Mr Neeson - has reportedly been at her son's hospital bedside.
The accident is the latest upset for Mr Neeson whose actress wife, Natasha Richardson, died in March 2009 at the age of 45 from head injuries following a ski accident in Canada.
In an interview with US show 60 Minutes in February,the 62-year-old Irishman said he was still struggling to come to terms with her loss five years on.
He said: "It was never real. It still kind of isn't. There's periods now in our New York residence when I hear the door opening - she would always drop the keys on the table and say hello.
"So any time I hear that door opening I still think I'm going to hear her. Grief hits you like a wave, you get this profound feeling of instability."
Sussex Police said there were no suspicious circumstances surrounding the incident involving Mr Sexton.
A police spokesman said: "Just after 4am on Sunday 22 June, a 31-year old Worthing man had climbed on to the top of a telephone kiosk outside Brighton Pier when he fell some 20 feet to the ground, sustaining a serious head injury.
"He was taken to the Royal Sussex County Hospital and was later transferred to Hurstwood Park Hospital in Haywards Heath where he is currently in a critical condition.
"The man had been enjoying a night out with friends, who were present when he fell, and the incident is not being treated as suspicious. There is no further police investigation."