Lately I've encountered several passages referring to people receiving a new divine name:
"What is your name?" the man asked. He answered, "Jacob." Then the man said, "You shall no longer be spoken of as Jacob, but as Israel, because you have contended with divine and human beings and have prevailed." --Genesis 32:28-29
"I will not let thee go," says Jacob in his night vision, "I will not let thee go, except thou bless me," and God answers, "What is thy name?" and when he hears it, corrects it: "Thou shalt no more be called Jacob, but Israel, for as a prince hast thou power with God and with men." Your name is perhaps little known to the world, and used, behind your back, in criticism and abuse; but there is a secret name by which God knows you, and it rings in his ears with a princely dignity; in all your insignificance, you are still a person to him; in all your weakness, you have still power in his counsels. When prayer seems difficult, try to remember that he knows you by such a name; that all the love and devotion with which you can pronounce the Holy Name of Jesus is returned, quite as individually and with far more steadfast purpose, in the unheard whisper with which he calls you, his own sheep, by name. Ronald Knox, Bread or Stone, pg. 12-13. (h/t Brandon)
God said to him: "You whose name is Jacob shall no longer be called Jacob, but Israel shall be your name." --Genesis 35:10
The nations shall see your vindication, and all the kings your glory; and you shall be called by a new name which the mouth of the LORD will give. --Isaiah 62:2
"I am created to do or to be something for which no one else is created: I have a place in God's counsels, in God's world, which no one else has. Whether I be rich or poor, despised or esteemed by man, God knows me and calls me by my name." Bl. John Henry Newman
Jesus said to him in reply, "Blessed are you, Simon son of Jonah. For flesh and blood have not revealed this to you, but my heavenly Father. And so I say to you, you are Peter, and upon this rock I will build my church, and the gates of the netherworld shall not prevail against it." --Matt. 16:17-18.Jacob fights with God, then answers the question about his name. Peter tells Jesus who He is, then Jesus tells Peter who he is. Both involve answering direct questions from God about identity. And both involve boldness. Jacob wrestles with God and prevails. Peter asserts that Jesus is the Messiah, even when others are more circumspect.
This private name in the mind of God doesn't seem to be something one can discover; rather, it seems that it must be revealed by God. But it also seems that He wants to reveal it.
Even if we don't consciously know our secret name, we must be able to respond to it, or else God wouldn't call us by it.