"To choose a successor" means to select and designate someone to take over a particular position, role, or job after the current person holding that position leaves or retires. Full definition
Instead of choosing a successor just like you, choose one to meet the new needs of the enterprise. (entrepreneur.com)
With a mayoral election to choose her successor coming this fall, her deep pockets could prove an asset to her allies. (syracuse.com)
Sir Alex Ferguson played a huge part in choosing his successor for Manchester United previously, and you would expect the Arsenal boss to also merit the same decision. (justarsenal.com)