Exorcism
In Catholic Christianity, exorcisms are performed in the name of Jesus Christ. A distinction is made between a formal exorcism, which can only be conducted by a priest during a baptism or with the permission of a bishop, and "prayers of deliverance" which can be said by anyone.
The Catholic rite for a formal exorcism, called a "Major Exorcism", is given in Section 11 of the Rituale Romanum. The Ritual lists guidelines for conducting an exorcism, and for determining when a formal exorcism is required. Priests are instructed to carefully determine that the nature of the affliction is not actually a psychological or physical illness before proceeding.
The Cairn on the Headland
Odin being in Howard's interpretation a demon, his banishment at the climax of the story is an effect an exorcism, and it includes many elements of an exorcism as depicted in Christian tradition. Holding out a crucifix is commonly mentioned as an effective method of confronting a demon. According to Christian theology - specifically, Catholic theology - one need not be a clergyman in order to perform an exorcism. However, if one is not especially holy oneself, the chances of overcoming the demon are much increased if one has a holy relic connected to a venerated Saint, thus being able to draw on that Saint's holiness. In all that, O'Brien follows on the well-established rules for carrying out an exorcism - and it works.