He would not be naming his successor, he would be naming a caretaker. Whomever he names will only be President for four years minus however long it takes to be confirmed by Congress, at which point he will either be reelected, or voted out of office. Cheney goes on TV, "In light of my health, I am not in a position to serve a four year term, therefore I will resign my office as soon as (Insert name here) is sworn in to the office of the Vice Presidency. Good night, and good luck."
He would be hailed as a hero for putting the interest of the nation ahead of his desire for power. He would not be criticized for at least ten years, and would take some lobbying job like every other Ex President. Oh, the Constitution by definition has no spirit. Spirits can only be possessed by living, or at least self aware beings. Any writ of Habeas Corpus would be shot down on appeal, and whomever brought the case would be seen as a vindictive ass for harassing a retired survivor of four heart attacks.
What would be the goal of the lawsuit? "I hate Cheney, so Cheney must be President for four years"?