The word rest in this quote from Hawthorne's story refers to the minister's impending death and subsequent time in the afterlife--it's similar to our maxim, "rest in peace."
Throughout his ministry to his parishioners, Minister Hooper sought to provide them with comfort and rest and to prepare them to go to their final "rest" with clear consciences. What is interesting in this story, though, is that the townspeople bury the minister with his black veil still on and view any attempt to remove it as hindering his "rest"--which is strange since there is supposed to be no sin (therefore, no need for the veil) in Heaven. The townspeople's choice is also ironic because the minister supposedly wore the veil to teach his people a spiritual lesson, but in the end, they attach a superstition and reverence to the veil which undermines the Hooper's motivation.
The phrase "it was now good Father Hooper's turn to rest" means that Mr. Hooper is about to die. In the Puritan-Christian religion, the souls of the righteous, such as Mr. Hooper, go to heaven after death, a place of eternal peace and rest.
Mr. Hooper had spent a long life ministering to the people in his congregation. In fact, "as years wore on...he acquired a name throughout the New England churches, and they called him Father Hooper." By virtue of the mysterious black veil he insisted on wearing since the time he was a young man, "he became a man of awful power, over souls that were in agony for sin," and "dying sinners cried aloud for Mr. Hooper, and would not yield their breath till he appeared." Father Hooper worked hard during his life, consoling many an individual in their last hours, and helping them to confess their sins and find forgiveness and peace before they breathed their last. Now he himself has grown old, and is lying on his deathbed. It is his turn to die; his earthly trials and travails are almost done, and he can look forward to entering heaven, that place of peace and rest.
Father Hooper had worn the black veil over his face for many years. He felt compelled to wear the veil for he had hidden sins that he himself could not feel repentant of. He wore the black veil to let his congregation as well as himself be reminded that each individual has sins within themselves that they try to hide from others.
When Father Hooper grows old it is the natural order of life that he would die. The phase means "it was now good Father Hooper's turn to rest," means that he can now finally die without the veil. However, when it comes time for Father Hooper to remove the veil and look upon his face, he can not do it. He still feels that he is sinful.