Who’s names are in the book of life?

Explain the Book of Life.  He that overcometh, the same shall be clothed in white raiment, and I will not blot out his name out of the book of life, but I will confess his name before my father, and before his angels.  Is this talking about Israel?  Rev 3:5.

When God is dealing with Israel (in the past or future), since Israel was in covenant relationship with God, a child was born into this covenant relationship (especially when they were circumcised at 8 days). Therefore their names were automatically entered into the book of life. The indications from Moses, when they worshipped the golden calf, is that their names would then be taken out of the book (Exodus 32:30-33). Since we are a new creature, our names did not go into the book of life until we believed the gospel and where they can never be taken out (Philippians 4:3). Interestingly, the passage in Revelation 17:8 where it speaks of those whose names never were in this book, is most likely speaking of Gentiles whose names never got in. Your passage, Revelation 3:5 is certainly referring to the Nation of Israel, but especially the believing remnant whose names will not be blotted out. These are those who will endure faithful through the great tribulation and overcome the hour of temptation.