Murray spent three years at the club between 2010 and 2013 before leaving to return to Oldham Athletic where he became first-team coach.
He heads back to a club where he was voted Player of the Year in 2011-12 and will embark on the first managerial role of his career.
His predecessor Cooper left the club earlier this month after a 3-0 defeat to Carlisle United - Hartlepool's 15th defeat in their last 21 competitive games.
Since Cooper's departure, Hartlepool have won one of their four games - beating Exeter City 2-1 at St James Park.
Murray's first game in charge comes on Saturday as they make the long trip south to face Cambridge United at the Abbey Stadium.
Willie Donachie will serve as Murray's assistant with Sam Collins continuing as first/reserve-team coach and Stephen Pears remaining goalkeeping coach.