Technically speaking it's by birthright, but it's a little more complicated than that.
Anyone who is in line to the throne is required to undertake extensive and exhaustive training for the task, starting as a child. There are various trials and tests they have to complete. They have to prove that they have both the technical ability to rule, have a deep understanding of the issues and tasks they will be faced with, as well as being intelligent, physically fit, and of strong moral character. Only after they have shown all of these are they eligible for the throne.
When the previous monarch dies or is otherwise removed from the throne, the Third House will choose a new one based on the available candidates. Usually this goes to the direct next in line, though it is their right to choose someone else if they feel that person is more appropriate.
Those in line for the throne are not allowed to abdicate the role unless they are directly the next in line, though informally a strong desire to not lead would probably be taken into consideration when choosing the next leader. This is to avoid issues of people abdicating so that they don't have to undergo the training. If the next in line wishes to abdicate, they must request this from the Third House, who have the right to refuse them. Abdication also results in the loss of your title and the privileges of royalty.
All of this is in an attempt to make sure that the person leading things actually knows what they're doing.
Although it has never occurred in practice, if an event were to happen where there were no eligible candidates available for the throne, the Third House would be required to take emergency measures to find a new ruler, and provide them with ongoing training. The exact nature of these measures is not specified, and changes depending on the current Head of House.
