The elder's main job is to make sure the clan has enough resources (including people). Thus, they are responsible for food store management (and must make an appropriate allowance for Pup-perpetrated theft), the movement of people in committed relationships (almost always with those of another clan), and the coordination of resources within the broader context of the Stenza army (although in practical terms this is dealt with by the First Officer). In all inter-clan relationships before and after Unification, this required two elders to negotiate with each other and anyone else who was an interested party to the subject of those negotiations. This has been the basis of many clan alliances and is also the basic principle behind the Council of Elders, which helps the Leader make decisions and provides law enforcement. As members of the Council, elders are required to hear and render judgment on major criminal cases (such as that of T'zim-Sha), hear and discuss the Leader's policy decisions (whether their input is considered or not, although a smart person would listen to them), select a prospect's target for the Leadership Challenge, and welcome the new Leader when the challenge has been completed successfully (or, as needed, rinse and repeat).
There is an extreme amount of social clout attached to being the oldest living member of a clan. It indicates that one has managed to survive a lot, is likely to have served or otherwise experienced conflict, and has seen at least some history. Elders are considered to have a particular insight into "the way the world works", which they can bring to bear on matters of state.