For those of you who don't know, there are two Hebrew words used in the Bible to refer to God. One is "elohim", the other is "yahweh" (or Jehovah). The LDS church often refers to "Elohim" as God the Father, and "Jehovah" as God the Son. (Note this doesn't mean that the church necessarily teaches that the passages in the Bible where elohim is used is necessarily God the Father speaking, or vice verse.)

I think, however, that the church does not claim those are the actual names of the Father and of the Son like John is my name, but rather that they are merely the words we use to refer to the Father and the Son. For example, Jehovah/Yahweh means "I am" or something like that, and does not seem to be a regular name. I don't know if "elohim" has a secondary meaning as well.

James Talmage, an apostle in the church (now deceased), called the terms, "name-titles". I think that is an apt description.

From Articles of Faith by Talmage:

"The purport of these scriptures is to the effect that God the Eternal Father, whom we designate by the exalted name-title 'Elohim,'..."

From Jesus the Christ, also by Talmage:

"The name Elohim is of frequent occurrence in the Hebrew texts of the Old Testament, though it is not found in our English versions. In form the word is a Hebrew plural noun; but it connotes the plurality of excellence or intensity, rather than distinctively of number. It is expressive of supreme or absolute exaltation and power. Elohim, as understood and used in the restored Church of Jesus Christ, is the name-title of God the Eternal Father, whose firstborn Son in the spirit is Jehovah--the Only Begotten in the flesh, Jesus Christ."