Fun Fact: Joseph Smith once changed the name of the church to the "Church of the Latter Day Saints" in 1834 He completely took "Jesus Christ" out of the name!
The LDS Church did not always have the name "The Church of Jesus Christ of Latter-day Saints".
On April 6, 1830, LDS church leaders met together presumably to fulfill the requirements of the laws of the land, and they named their church consistent with the admonition of the Book of Mormon, "The Church of Christ."
On May 3, 1834, Joseph Smith, as moderator, made a motion to change the name of the church to, "The Church of the Latter Day Saints," which was adopted by unanimous vote (The Evening and Morning Star, vol. 2, no. 20, p. 160, May 1834; History of the Church 2:62). He took CHRIST right out of the name!
Four years later, on April 26, 1838, it was again changed to "The Church of Jesus Christ of Latter Day Saints" and has remained thus since. (The hyphen in 'Latter-day' was add about a century or so later to be grammatically correct).
The chronology of the name is:
1830: Church of Christ
1834: The Church of the Latter Day Saints
1838: The Church of Jesus Christ of Latter Day Saints
Why did Joseph change the Name of the Church after God already told him 4 years before what to name it? "In June, 1829, the Lord gave us the name by which we must call the church, being the same as He gave the Nephites. We obeyed His commandment, and called it The Church of Christ until 1834, when, through the influence of Sydney Rigdon, the name of the church was changed to The Church of the Latter Day Saints, dropping out the name of Christ entirely..." (An Address to All Believers In Christ Whitmer, p. 73). Now to be called the Church of Jesus Christ of Latter-day Saints (LDS).
The following photograph of the Kirtland temple still bears the name 'The Church of the Latter Day Saints'.