What is the meaning of "rest in peace" ("RIP")?

Interestingly, the word rest in the New Testament is something each believer in Jesus Christ can experience, this side of the grave! The writer to the Hebrews addresses the concept of rest in Chapter 4. He posits that only those believers in Jesus Christ who are faithfully obedient to the Lord will enter into rest.

In other words, the promise of entering into God's rest was applicable not only to the Israelites as they journeyed toward the land of promise, but it is also applicable to believers under the New Covenant, from Christ's resurrection in the first century to the 21st century and beyond.

That rest is experienced by faithfully-obedient believers in Jesus as a surcease of striving to earn God's favor. Through Christ's death, burial, and resurrection, he alone earned the right to offer that rest to all who believe. Believers experience that rest in God's undeserved and unearned favor. We call that grace.

As for "resting in peace" (frequently abbreviated RIP), that phrase is not biblical in origin. The surcease of striving--in a general sense--certainly ends for all people--believers and unbelievers alike--when they die, but for the unbelievers who failed to enter that rest while alive, the afterlife will be anything but peaceful. The parable of Lazarus and the rich man makes this abundantly clear.

“There was a rich man who was dressed in purple and fine linen and lived in luxury every day. 20 At his gate was laid a beggar named Lazarus, covered with sores 21 and longing to eat what fell from the rich man’s table. Even the dogs came and licked his sores. "The time came when the beggar died and the angels carried him to Abraham’s side. The rich man also died and was buried. In Hades, where he was in torment, he looked up and saw Abraham far away, with Lazarus by his side. So he called to him, ‘Father Abraham, have pity on me and send Lazarus to dip the tip of his finger in water and cool my tongue, because I am in agony in this fire.’

“But Abraham replied, ‘Son, remember that in your lifetime you received your good things, while Lazarus received bad things, but now he is comforted here and you are in agony. And besides all this, between us and you a great chasm has been set in place, so that those who want to go from here to you cannot, nor can anyone cross over from there to us.’ “He answered, ‘Then I beg you, father, send Lazarus to my family, for I have five brothers. Let him warn them, so that they will not also come to this place of torment.’

“Abraham replied, ‘They have Moses and the Prophets; let them listen to them.’ “‘No, father Abraham,’ he said, ‘but if someone from the dead goes to them, they will repent.’ “He said to him, ‘If they do not listen to Moses and the Prophets, they will not be convinced even if someone rises from the dead’” (Luke 16:19-31 NIV, my bolding).

The only Greek scripture verse that contains both "soul" and "peace" is:

Now may the God of peace Himself sanctify you completely; and may your whole spirit, soul, and body be preserved blameless at the coming of our Lord Jesus Christ.
— 1 Thessalonians 5:23

And even this one scripture has nothing to do with death.

"RIP" and "Rest In Peace" come from the Latin phrase "Requiescat In Pace", perhaps dating back to the 3^rd century.

Like many other expressions:

• God works in mysterious ways.
• Cleanliness is next to godliness.
• God helps those who help themselves.
• The Seven Deadly Sins.
• Love the sinner. Hate the sin.

it is not from the Bible.

The traditional use

The wikipedia article traces the phrase "requiescat in pace" ("may he/she rest in peace") to tombstone inscriptions as early as the 5th century.

Up to the Reformation, most Christian groups would offer prayers for the dead in connection with the belief of the communion of saints (part of the Apostle's creed) which refers to the spiritual union of the living and the dead but excluding the damned. See wikipedia on how different groups do this.

In Catholic funeral mass, the entrance antiphon has the following words as a prayer:

Eternal rest grant unto them, O Lord, and let perpetual light shine upon them.

An example of this text set to music is the Introit of Mozart's Requiem (see text, sample performance, minute 1:42). The music text (in Latin) is the same:

Requiem ætérnam dona eis, Dómine, et lux perpétua lúceat eis.

Is it Biblical to pray for the dead?

While I agree with the GotQuestions article Is it biblical to say ‘rest in peace’ (RIP) in regards to someone who has died? that it is NOT Biblical to say that all people who die WILL rest in peace, I agree with @jaredad7's comment that

RIP stands for requiescat in pace, which means may he/she/it rest in peace. Requiescat is in the subjunctive mood. This is a saying wishing someone to rest in peace, and that is biblical. We should hope and pray for everyone to rest in peace.

Christian groups that continue the early church's understanding of the communion of saints will say that the practice is Biblical, some appealing to a plausible interpretation that Onesiphorus whom Paul prayed on behalf of (2 Tim 1:16-18) had died. While some groups say that the dead's fate is sealed upon death (making our prayers for them useless), other groups pray for the saved to increase in love, as taught in the catechism in the 1979 Anglican Book of Common Prayer:

We pray for (the dead), because we still hold them in our love, and because we trust that in God's presence those who have chosen to serve him will grow in his love, until they see him as he is.

How about the dead's actual rest / peace?

What's the Biblical support that at least some believers will actually die in peace? I think Heb 4:1-3 makes it clear that believers who die in Christ can be said to enter God's rest:

God’s promise of entering his rest still stands, so we ought to tremble with fear that some of you might fail to experience it. For this good news—that God has prepared this rest—has been announced to us just as it was to them. But it did them no good because they didn’t share the faith of those who listened to God. For only we who believe can enter his rest. As for the others, God said,

“In my anger I took an oath: ‘They will never enter my place of rest,’” [Ps 95:11]

even though this rest has been ready since he made the world.

Especially for those who also suffer greatly because of war or because of defending Christ, it will also be meaningful to say that they now "rest in peace" as they have been freed from their earthly torments.