Jesus' Example for Us
Jesus calls The Father "Abba" and authorizes us to do so also
Jesus knew at the core of his being that he was the beloved Son (Lk 2:49, 3:22, 9:35, 20:13-14; Mt 17:5; Mk 9:7, 12:6-7; Jn 5:20), and he expressed this intimacy by using the affectionate Aramaic term, “Abba” (Mk 14:36), like our “Dad.” Mark 14:36 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), whenever the Greek “Father” (pater) is recorded in Jesus' prayer, we can infer he was 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 (Mt 23:9). Additional encouragement to use this term of endearment in our prayer life comes from the Apostle Paul: “The Spirit you received does not make you slaves, so that you live in fear again; rather, the Spirit you received brought about your adoption to sonship. And by him we cry, ‘Abba, Father.’” (Rom 8:15). “Because you are his sons, God sent the Spirit of his Son into our hearts, the Spirit who calls out, “Abba, Father” (Gal 4:6).
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), p. 78.
 Scot McKnight, The Jesus Creed (Brewster, MA: Paraclete Press, 2004), p. 25.
For further discussion see, Klaus Issler, Living into the Life of Jesus: The Formation of Christian Character (InterVarsity, 2012).