See: D&C 132:19
Those who devote themselves to the Lord earn for themselves the privilege of having their names recorded in the Lamb’s book of life. This sacred “book of remembrance” (v. 16) is kept in heaven and contains the names of the faithful children of Father in Heaven, or, in other words, those who are His precious jewels. (See D&C 128:6-7; Psalm 69:28; Revelation 3:5; 21:27.) They are those who will inherit eternal life, for this book contains “the names of the sanctified, even them of the celestial world” (D&C 88:2). Those whose names are written there and who afterwards return to sinful ways will have their names blotted from the book (see Revelation 13:8; 17:8; 22:19).
The book of life is the Lord’s book of remembrance, not unlike records that He has commanded the Saints to keep in all ages of the world. Elder McConkie explained:
“Adam kept a written account of his faithful descendants in which he recorded their faith and works, their righteousness and devotion, their revelations and visions, and their adherence to the revealed plan of salvation. To signify the importance of honoring our worthy ancestors and of hearkening to the great truths revealed to them, Adam called his record a book of remembrance. It was prepared ‘according to the pattern given by the finger of God.’ (Moses 6:4-6, 46.)
“Similar records have been kept by the saints in all ages. (Mal. 3:16-17; 3 Ne. 24:15-16.) Many of our present scriptures have come down to us because they were first written by prophets who were following Adam’s pattern of keeping a book of remembrance. The Church keeps similar records today (D&C 85) and urges its members to keep their own personal and family books of remembrance.” (Mormon Doctrine, p. 100; Doctrinal New Testament Commentary, Vol. III, p. 455) — Old Testament Student Manual, 1 Kings – Malachi, p. 355