Many people assume that we call ourselves Latter-day Saints because we think we're holier than everybody else. Not so. The term "saint" is what the Lord calls members of His Church. It reminds us that we are called to live His laws, but we are utterly imperfect and have no monopoly on righteousness.
The term "saints" in our name is just one little part of the Restoration of the Church of Jesus Christ, bringing back some of the ancient ways that the Lord has seen fit to apply in for our day as well.
I suppose that if Joseph Smith were just making everything up by absorbing elements in his environment, he may not have used the terms "saints" as he did. In the modern mainstream Christian world, it has generally taken on a different meaning, referring to rare individuals rather than the members of the Church in general. Indeed, the entry for "saint" in the 1828 Webster's Dictionary gives definitions that do not suggest Joseph Smith's usage:
SAINT, n. [L. sanctus.]I truly believe that there has been a divine restoration of the Church of Jesus Christ, and to me it only makes sense that this tiny little aspect of the original Church would be one of the many components that were restored. I say this not as any kind of serious evidence for the Restoration (turn to the Book of Mormon for that!), but by way of sharing some thoughts about that puzzling term in the name of our church.
1. A person sanctified; a holy or godly person; one eminent for piety and virtue. It is particularly applied to the apostles and other holy persons mentioned in Scripture. A hypocrite may imitate a saint. Ps. 16.
2. One of the blessed in heaven. Rev. 18.
3. The holy angels are called saint. Deut. 33. Jude 14.
4. One canonized by the church of Rome.