A book of remembrance is a record God keeps. It lists the names of people who honor and obey him.
The book is mentioned one time in the Old Testament (Malachi 3:16). The prophet Malachi was speaking during a time when many people were turning away from God. Evil people were growing rich and proud. It looked like God was not doing anything about it.
Malachi said that God was still watching. God would remember those who stayed faithful. He would one day show the difference between the righteous and the wicked.