I heard a preacher say that all people's names are originally in the book of life. However, those who do not accept Christ are blotted out of the book. There are quite a few verses that seem to support this view.
Exd 32:31 And Moses returned unto the LORD, and said, Oh, this people have sinned a great sin, and have made them gods of gold.
Exd 32:32 Yet now, if thou wilt forgive their sin; and if not, blot me, I pray thee, out of thy book which thou hast written.
Exd 32:33 And the LORD said unto Moses, Whosoever hath sinned against me, him will I blot out of my book.
Rev 3:5 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.
Believeing means to cease from works and trust or rely completely ON Christ. I compare it to jumping out of an airplane with a parachute. In this situation, you place yourself in a completely helpless state. If the parachute fails, you will plunge to your death.
I trust completely in Jesus to get me to heaven. I am glad and grateful that I do not have to earn salvation through works. I have not gotten too far in this world through my works, I know for certain that I would fail if I tried to earn heaven.
Jhn 6:28 Then said they unto him, What shall we do, that we might work the works of God?
Jhn 6:29 Jesus answered and said unto them, This is the work of God, that ye believe on him whom he hath sent.
As for election, I believe that God chose or elected from the beginning to save those who would trust in Jesus.
I also believe that God knew from the beginning who would choose to trust in Jesus. I used to ask myself why God would create a person whom he knew would reject Christ.
Then I realized that it must be so. We must have free choice. If not, we would be mere robots. Love involves free will. It would not be love if we had no choice. No, God wants those who choose him from their own free will. And God must also give the choice to perish to those who reject him.
If a person perishes, it is their own fault and personal choice.