Becoming a king in Medieval Ireland

Who is eligible to rule?
Every member of a king's derbfine, or extended family within four removals, was eligible to inherit his kingship. This created some problems for how to decide who the next king would be, since as often as not, the previous king was deceased at the time of the new king's succession, and couldn't name his chosen heir of the plethora of available relations. The kingship didn't usually pass directly from father to son either, since fathers often died too quickly, leaving sons too young to effectively rule as kings.
In order to narrow down the candidates for kingship, it was necessary for the individual to have rígdomna. Rígdomna can be defined as the attributes necessary for kingship, or the stuff that makes a king, and comprises wisdom, bravery, and even physical appearance. Only the members of the derbfine who exhibited high amounts of rigdomna were considered likely successors.

How is a king chosen?
Once the field of candidates was narrowed, the actual decision needed to be made. This could be done in a number of ways: tanieserig, banfeis, bull feast, and ordaination. A tanieserig is a person who is next in line for the throne, and cannot be displaced from this position; a candidate chosen by the current king, or the only member of the king's derbfine, would be examples of a tanieserig.

Banfeis was a common act of gaining the kingship in the literature of the time, which may or may not have been an actual practice. It referred to sleeping with a woman, particularly but not only in the marital bed. The Irish people of the time believed that lands themselves were personified by women, so the thinking was if that mystical woman gave herself to you, you would receive not just rights to her, but to the land she represented as well. This is most clearly shown in the story of Niall. Naill was the youngest of five sons to a king, and born to a different mother, so was something of an outcast. One day the five of them were hunting in the woods, but were very thirsty. One of the older brothers found a well guarded by an ugly hag, who said that he could draw from the well only if he kissed her, with kissing here referring to a more intimate interaction. He refused, as did each of his brothers, except for Naill. When Naill lied down with her, she transformed into a beautiful woman, who identified herself as Sovereignty and gave him seniority over his brothers, and thus the first right to kingship for him and his descendants.
A bull feast is when a man would slaughter a bull, eat till he was full and then sleep. This man needed to have the gift of prophesy, so was a poet, or in many cases a druid. While sleeping, other druids murmured incantations of truth over him and he would dream of the man would should be the next king. This is not to say he would dream of a particular person, but rather a generic person doing something unusual, such as walking down the road to town wearing nothing but their sling. For more information on druids please go here.

Ordaination was the only Christian method of the four, which occurred only in the later stages of early medieval Ireland, after Christianity became a pervasive part of Irish life. Ordaination occurred when the church selected the person whom they considered to be the most eligible for rule, then granted that person the kingship, endorsing their leadership with supposed divine acceptance. For more information on the clergy in medieval Ireland please go here.

Stories about men becoming kings:
Niall, King of Ireland (A.D. 379 - 405)
Cormac, son of Art
Conare Mar, King of Ulaid
Ailill, husband of Medb and King of Connacht