Why is it called The Church of Jesus Christ of Latter-day Saints?

The Church is so named because:

1. Mormons believe that this church is the same church that Jesus Christ established when he was on the earth, same organization and everything. It's his church and his doctrine. Therefore, the Church should bear his name.
2. Mormons believe that after Jesus died, the Church began to lose the purity of the teachings of Jesus Christ (the Gospel). The Church started teaching the philosophies of men mixed with the Gospel, and so it stopped being God's true church. Christianity became divided and corrupt, and it remained in this confused state until the world was ready to receive the pure Gospel again. In New York in 1830, Jesus Christ restored his church through the prophet Joseph Smith. (See Joseph Smith (mormon.org)) Joseph received a revelation that God wanted the name of the church to be the Church of Jesus Christ of Latter-day Saints. So that's what he called it. 
3. Members of the Church are called saints because that's what members of the ancient church were called. Members today are called latter-day saints because we are living in the Latter Days, the period of time that takes place just before Jesus' second coming.
4. The Church is often referred to as the Mormon Church, and members are commonly called Mormons, because they believe in the Book of Mormon as well as in the Holy Bible. The Book of Mormon is another witness of Jesus Christ. It recounts the history of two ancient-American nations, the Nephites and the Lamanites. The Book testifies of Jesus Christ and teaches his Gospel. (See The Book of Mormon (mormon.org))

The Church could have been given this name: The Church That's the Same as Christ's Ancient Church Whose Members Are Living in the Time Just before the Savior Comes Again. After that, the actual name of the church doesn't seem too long, now does it?

See also these Mormon.org articles: The Restoration of Jesus Christ's Church, Christ's Church

The "Seminary Answers"

Education is an important principle to Mormons. That's why church is 3 hours long, and the Church publishes all kinds of books and materials to help teachers teach and students learn. There are also many classes that the Church provides (such as Sunday S\chool, financial seminars, leadership training, etc.), and seminary is one of them. Seminary is a four-year class for high school students. Each year the class focuses on either the Book of Mormon, the Old Testament and the Pearl of Great Price, the New Testament, or the Doctrine and Covenants and church history. The class is typically held five days a week during the regular school year, which can get pretty tedious for a busy, stressed-out teen. (I would know. I lived it.)

The terms "Seminary answers" and "Sunday School answers" have come to mean the answers to a question that pretty much everyone in the room knows but will not say because it's so obvious. For example, a Sunday School teacher might ask, "How can we come closer to God? Any thoughts?" The Seminary answers are: 1) Pray. 2) Read your scriptures. 3) Go to church. 4) Repent. 5) Develop your faith in Jesus Christ (by doing what he says and such).

Those four answers are the answers to SO MANY QUESTIONS that teachers will ask, probably because they are pretty much the answers to life. Because these answers are so universal, and because the concepts in the answers are taught day after day, after day, after day, after day.....in seminary, and week after week in Sunday School, they are called the "Seminary answers" or the "Sunday School answers."

Or at least that how I would explain it.