It seems to me that the you need non-combat mechanics for most of the same reasons you want combat mechanics. Depending on your tolerance for complexity, you may want:
1) A system that allows the players (and DM) to track how close or far away they are from success or failure. This provides a feeling of progress and, while close to failure, a sense of excitement. (Incidentally, this is one of the reasons I subscribe to PCat's theory of making the skill challenge explicit and letting the PCs track successes and failures. It is also why a visible "bloodied" condition is such a good idea.)
2) Mechanics that allow different characters to interact with the challenge in different ways. Giving players their areas of expertise (relative to the other PCs) differentiates the characters and gives them different ways and opportunities to shine.
3) A system that provides opportunities for interesting decisions and tactical reasoning. In a skill challenge, there is the opportunity for creativity in how to apply your better skills in (non-obvious) ways that are appropriate to the situation. In Dogs in the Vineyard, there is the similar strategy of bringing in traits and relationships into the conflict, as well as the decision as to whether or not to accept the more severe consequences of escalation.
4) An excuse and a mechanism to get everyone at the table engaged in solving the problem. Some systems are better at this than others, but a more complicated system can provide more chances for every play to contribute input.
The one thing I'll note is that social challenges may not need a rules mechanism, because the player's ability to act out the interaction can itself be the source of these characteristics.
-KS