By his great mercy he has given us a new birth...
A new birth signifies a new life. Jesus told Nicodemus "No one can see the kingdom of God without being born from above" (Jn 3:3). Does Peter have this dialogue in his mind when he pens his letter? Maybe so.
Later in his letter, Peter defines what he means by "new birth". You have been born anew, not of perishable but of imperishable seed, through the living and enduring word of God (1:23)
Peter states this birth is into a living hope (1:4) Have you ever observed someone without hope? The Apostle Paul mentions non-believers as having no hope (Eph 2:12). Peter could have begun his letter in many ways. However, one of the first things he offers is hope. The Christian must understand what Christ has done for them - even in the midst of persecution.
One other thing, when it comes to new birth, John writes, "...for whoever is born of God conquers the world, our faith" (1 Jn 5:4). That's the hope Peter is speaking of.
During this Easter season, we remember what Christ Jesus has done for us. We remember the thirty-nine lashes, the mocking, the sin placed upon Him, the gruesome death, and the burial. And then we remember the tomb being empty, Jesus appearing to his followers, the restoration of Peter (among others), and the hope He gives. That's the hope Peter speaks of. That's the hope that we must continue to attempt to understand.