Jesus calls The Father "Abba" and authorizes us to do so as well.
Jesus knew at his core that he was the beloved Son (Luke 2:49, 3:22, 20:13-14; Mark 9:7, 12:6-7; John 5:20), and he expressed this intimacy by using the affectionate Aramaic term, “Abba” (Mark 14:36), like our “Dad.” This is the only record of this word on Jesus’ lips. Yet, since Jesus mainly spoke Aramaic (Mark 5:41, 7:34, 15:34; John 1:42; Acts 26:14), we can infer that every use of the Greek “Father” (pater), that Jesus was actually speaking the Aramaic, “Abba.” Joachim Jeremias, in his classic study on this topic, notes that “Jesus talks to his Father as naturally, as intimately and with the same sense of security as a child talks to his father.” 
Jesus introduced a new way of addressing God, with familial familiarity, and encourages us to do so as well, a special term to be reserved for the Father alone (Matt 23:9). Additional encouragement to use this term of endearment in our prayer life comes from the Apostle Paul: “For you did not receive a spirit of slavery to fall back into fear, but you have received a spirit of adoption. When we cry, ‘Abba! Father!’” (Rom 8:15 NRSV). “And because you are children, God sent the Spirit of his Son into our hearts, crying, “Abba! Father!” (Gal 4:6 NRSV).
A helpful summary on this matter comes from New Testament scholar Scot McKnight:
“Jesus is decidedly lopsided when it comes to names for God: every prayer of Jesus recorded in the Gospels begins with ‘Abba, Father’ except the famous ‘My God, My God, why have you forsaken me?’ utterance from the cross. . . . What Jesus wants to evoke with the name Abba is God’s unconditional, unlimited, and unwavering love for his people. In this name for God we are standing face-to-face with the very presence of spiritual formation: God loves us and we are his children.” 
 Joachim Jeremias, The Prayers of Jesus (Philadelphia, PA: Fortress, 1967), 78.
 Scot McKnight, The Jesus Creed (Brewster, MA: Paraclete Press, 2004), 25.