Angie's uncle felt that calling God "Daddy" is too informal; we have to "maintain a level of reverence for Holy God" and should address God as Abba, Father which is what is written in the scriptures. This also seems to be the position of many theologians and scholars when I researched the meaning of "Abba".
The term Abba appears only 3 times in the bible:
Mark 14:36 (NLT)
“Abba, Father,” he cried out, “everything is possible for you. Please take this cup of suffering away from me. Yet I want your will to be done, not mine.”
Romans 8:15 (NIV)
For you did not receive a spirit that makes you a slave again to fear, but you received the Spirit of sonship.[a] And by him we cry, "Abba,[b] Father."
Galatians 4:6 (NLT)
And because we are his children, God has sent the Spirit of his Son into our hearts, prompting us to call out, “Abba, Father.”
In Mark, Jesus was the one crying out Abba. In Romans and Galatians, Paul was the one who introduced the term Abba. In all 3 instances, it is about the son calling out to the father.
Paul was telling the believers that calling God "Abba" was a privilege that they never had before. Through Jesus' finished work on the cross, we are now reconciled to the Father; we are now adopted as sons, giving us the privilege of calling Him "Abba".
Before Jesus' death, burial and resurrection, only Jesus himself called the Father "Abba". In Matthew 28:10, after Jesus rose from the dead, He appeared to the two Mary(s) and said "Do not be afraid. Go and tell my brothers to go to Galilee; there they will see me." Notice that Jesus now called his disciples "my brothers" i.e. they are now sons of God too and just like Him they also have the privilege of calling God "Abba".
Isn't it interesting that the term Abba appears only in the New Testament? This is because we can call God "Abba" only when we are in Christ.
Romans 8:15 says you did not receive a spirit that makes you a slave again to fear. The word "again" implies that you were previously a slave to fear before: when the Israelites stood at the foot of Mount Sinai to receive the Law.
Exodus 20:18-20 (NIV)
18 When the people saw the thunder and lightning and heard the trumpet and saw the mountain in smoke, they trembled with fear. They stayed at a distance 19 and said to Moses, "Speak to us yourself and we will listen. But do not have God speak to us or we will die."
20 Moses said to the people, "Do not be afraid. God has come to test you, so that the fear of God will be with you to keep you from sinning."
Who is the best person to reveal God, the Father to us other than Jesus himself? John 10:15 says "just as the Father knows me and I know the Father—and I lay down my life for the sheep."
There was no need for Jesus to introduce the fear of God to the Jews because they already know it under the law. Jesus came to reveal the love of God, the grace of God.
John 1:14,17-18 (NKJV)
14 And the Word became flesh and dwelt among us, and we beheld His glory, the glory as of the only begotten of the Father, full of grace and truth.17 For the law was given through Moses, but grace and truth came through Jesus Christ. 18 No one has seen God at any time. The only begotten Son,[f] who is in the bosom of the Father, He has declared Him.
Notice that verse 18 says that Jesus is in the bosom of the Father: this speaks of intimacy, not a distant reverence. This is mirrored in John 13:23 "Now there was leaning on Jesus’ bosom one of His disciples, whom Jesus loved."
Jesus wants to reveal to us that God, the Father craves intimacy with us who are His children. Jesus did this through the parable of the prodigal son (Luke15:11-31). We call it the parable of the prodigal son but in actual fact the parable is all about the love of the father.
The parable illustrates the unfailing love of the father. When the son demanded his inheritance, the father could have thrown him out of the house without a single cent but he did not. When the son came back penniless, the father could have waited for the son to kneel down before him at the house but he did not.
Instead, "while he was still a long way off, his father saw him coming. Filled with love and compassion, he ran to his son, embraced him, and kissed him."
Jesus has just shown us the Father who wants to embrace us and kiss us, the Father who wants us in His bosom: why do we want to remain distant and reverent?
As for me, I want to hug my heavenly father with all my might and and cry out "Daddy! Daddy!"