Several times in the Bible a person's name is repeated, often when God is speaking to them. Repetition is usually used for emphasis, and repeating a name implies intimacy with the person.
When Abraham is about to sacrifice his son Isaac, God stops him:
But the angel of the Lord called to him from heaven and said, “Abraham, Abraham!” And he said, “Here I am.” He said, “Do not lay your hand on the boy or do anything to him, for now I know that you fear God, seeing you have not withheld your son, your only son, from me.”
(Genesis 22:11-2)
God periodically revealed himself to Jacob; this is the final time he does so:
And God spoke to Israel in visions of the night and said, “Jacob, Jacob.” And he said, “Here I am.” Then he said, “I am God, the God of your father. Do not be afraid to go down to Egypt, for there I will make you into a great nation...
(Genesis 46:2-3)
When God appears to Moses in the burning bush, he also repeats Moses' name:
“When the Lord saw that he turned aside to see, God called to him out of the bush, “Moses, Moses!” And he said, “Here I am.”
(Exodus 3:4)
Samuel's call is also given at night:
And the Lord came and stood, calling as at other times, “Samuel! Samuel!” And Samuel said, “Speak, for your servant hears.”
(1 Samuel 3:10)
As we come to the New Testament, this repetition is mostly used by Jesus. When Martha sees that her sister Mary is sitting at Jesus' feet rather than helping her, she wants Jesus to intervene:
But the Lord answered her, “Martha, Martha, you are anxious and troubled about many things...”
(Luke 10:41)
Jesus speaks to Peter (who was originally called Simon) and predicts Peter's denial and subsequent restoration:
“Simon, Simon, behold, Satan demanded to have you, that he might sift you like wheat, but I have prayed for you that your faith may not fail. And when you have turned again, strengthen your brothers.”
(Luke 22:31-32)
Then in the Book of Acts, Jesus speaks to Paul (originally called Saul) on the Damascus Road:
And falling to the ground, he heard a voice saying to him, “Saul, Saul, why are you persecuting me?” And he said, “Who are you, Lord?” And he said, “I am Jesus, whom you are persecuting...”
(Acts 9:4-5)
It's not just people who are addressed in this way; Jesus uses the double name when he weeps over Jerusalem:
“O Jerusalem, Jerusalem, the city that kills the prophets and stones those who are sent to it! How often would I have gathered your children together as a hen gathers her brood under her wings, and you were not willing!”
(Matthew 23:37)
And Jesus repeats God's name when he prays on the cross, quoting Psalm 22:1:
And about the ninth hour Jesus cried out with a loud voice, saying, “Eli, Eli, lema sabachthani?” that is, “My God, my God, why have you forsaken me?”
(Matthew 27:46)
When we look at all these occurrences, we see that in repeating people's names in particular circumstances, God shows us that he is a relational God. He speaks to get our attention, and he speaks to our hearts. He is the God who pursues intimacy with us.
Finally, some pretend to have a deep relationship with God, but they don't actually know him. This pattern helps us to better understand Jesus' words in the Sermon on the Mount:
Not everyone who says to me, “Lord, Lord,” will enter the kingdom of heaven, but the one who does the will of my Father who is in heaven.
(Matthew 7:21)