Rev 3:5 He who overcomes shall be clothed in white garments, and I will not blot out his name from the Book of Life; but I will confess his name before My Father and before His angels.
Believers in Christ are over comers in General and when we die we will reach the paradise also:
1Jn 4:4 You are of God, little children, and have overcome them, because He who is in you is greater than he who is in the world.
1Jn 5:4 For whatever is born of God overcomes the world. And this is the victory that has overcome the world—our faith.
1Jn 5:5 Who is he who overcomes the world, but he who believes that Jesus is the Son of God?
But it is related to rewards also, the believers who triumphant over trials and tribulations and also worked for the Glory of God.
Rom 8:37 Yet in all these things we are more than conquerors through Him who loved us.
2Co 2:14 Now thanks be to God who always leads us in triumph in Christ, and through us diffuses the fragrance of His knowledge in every place.
Second death means eternal separation from God and going to lake of fire, it is for the unbelievers.
Rev 20:14 Then Death and Hades were cast into the lake of fire. This is the second death.
Rev 20:15 And anyone not found written in the Book of Life was cast into the lake of fire.
Second death has no power over the believers in Christ and they do not have any judgement or condemnation.
Rev 20:6 Blessed and holy is he who has part in the first resurrection. Over such the second death has no power, but they shall be priests of God and of Christ, and shall reign with Him a thousand years.
Rom 8:1 There is therefore now no condemnation to those who are in Christ Jesus, who do not walk according to the flesh, but according to the Spirit.
Joh 3:18 "He who believes in Him is not condemned; but he who does not believe is condemned already, because he has not believed in the name of the only begotten Son of God.
Joh 5:24 "Most assuredly, I say to you, he who hears My word and believes in Him who sent Me has everlasting life, and shall not come into judgment, but has passed from death into life.
In Revelation 3:5, the over comers (who triumphant over the trials and tribulation and work for the glory of God) has not only clothed with white garments but giving the assurance that their name will not be blotted out from the book of life.
Please refer to Rev 2: 10 & 11
Rev 2:10 Do not fear any of those things which you are about to suffer. Indeed, the devil is about to throw some of you into prison, that you may be tested, and you will have tribulation ten days. Be faithful until death, and I will give you the crown of life.
Rev 2:11 "He who has an ear, let him hear what the Spirit says to the churches. He who overcomes shall not be hurt by the second death." '
A believer can loose his reward like Crown of life or any other rewards if the person is not faithful/cannot endure trials or backslidden etc.
“Be faithful until death, and I will give you the crown of life”
“He who overcome shall not be hurt by the second death”
Here persecution/tribulation is there it can go up to death, as martyrdom, even if one become martyrdom (physical death) by a believer in Christ, the person will not hurt by the second death.