Abba in the Aramaic sections of the Old Testament is the standard Aramaic word for father, for example “our fathers” in Ezra 4:15, 5:12, Daniel 2:23, or in the singular:
Belshazzar, whiles he tasted the wine, commanded to bring the golden and silver vessels which his father Nebuchadnezzar had taken out of the temple which was in Jerusalem; that the king, and his princes, his wives, and his concubines, might drink therein. (Daniel 5:2 KJV)
Despite this, various popular writers claim that during the time of Jesus the Aramaic Abba was childish language equivalent to Papa, or Daddy. This idea goes back to early sources such as Chrysostom who, in his own childhood, noted this as an Aramaic form of child’s address for a father.
Much of modern credence for this idea comes not from reading Chrysostom, but from Joachim Jeremias who has an extensive section on ‘Abba in the Prayers of Jesus’ in his influential and often reliable book ‘Jesus and the Message of the New Testament’ (pages 68-70). This is followed directly by further development of the idea in the next section ‘The Fatherhood of God in the Gospels’ leading on from this “childlike address to God” (pp 70-71). Jeremias notes that this usage “, which surely originated from the idiom of the small child, had vastly extended its meaning in Palestinian Aramaic”, but nevertheless focuses on the child’s usage.
Notable pushback against Jeremias’ reading comes in James Barr “‘Abbā Isn’t ‘Daddy,'” Journal of Theological Studies Vol39/No1 April 1988 pp. 28-47 (20 pages) Oxford University Press
If the New Testament writers had been conscious of the nuance ‘Daddy’ they could easily have expressed themselves so; but in fact they were well aware that the nuance is not that of ‘Daddy’ but of ‘father’.” Thus, to sum up this section, the semantics of ‘abbā itself, the usage of the Targum, and the choice of vocabulary in the New Testament all agree in supporting the nuance ‘father’ than the nuance ‘Daddy’.” …
Conclusions:
“(a) It is fair to say that ‘abbā in Jesus’ time belonged to a familiar or colloquial register of language, as distinct from more formal and ceremonious language, though it would be unwise, in view of the usage of the Targum, to press this too far. But in any case it was not a childish expression comparable with “Daddy”: it was a more solemn, responsible, adult address to a Father.”
… (also found in Bible and Interpretation: The Collected Essays of James Barr. Page 271 and 279)
A number of other scholars have written similarly, with similar conclusions, that this is an informal, but still respectful, adult address not equivalent to English “daddy” or Greek “papa”.