Why do Mormons use the term "elder" for their missionaries? Most of the Mormon missionaries I've encountered are young men doing a couple year stint. They are typically identified as "elders", but my understanding is that this designation is only temporary during their time of service.

How is the LDS use of this term similar or different than the office of elder in mainstream Protestant circles? Does their designation as elders give them any ruling authority over the church body or is their role strictly to work with non members?
The term "Elder" actually has two (related) meanings among Latter-Day Saints: first, it's an office in the Priesthood, which all worthy male members can attain.  It doesn't imply any leadership, just certain duties, responsibilities, and privileges associated with being an Elder in the Priesthood.  All male missionaries must be found worthy to be ordained an Elder before leaving on their mission.

But not all members who have been ordained an Elder go by the title of "Elder".  That's used for people who are dedicated full-time to the ministry.  As the Latter-Day Saints have a lay clergy, this is a fairly small group.  It obviously includes the missionaries, and General Authorities of the Church (the Apostles and Seventies who preside over the Church as a whole) are also referred to as "Elder [last name]."

As for a missionary's authority in the Church, they have no inherent position of leadership as part of their calling, though missionaries assigned to areas where the Church is still small may find themselves called to serve in the leadership of the local branch while they're there.  But their primary calling is to teach the Gospel, not to administer in the affairs of the Church.