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Cyclopedia of Biblical, Theological and Ecclesiastical Literature
(‘Αββᾶ, אִבָּא ) is the Hebrew word אָב, father, under a form (the "emphatic" or definite state — the father) peculiar to the Chaldee idiom (Mark 14:36; Romans 8:15; Galatians 4:6).
1. As such, it was doubtless in common use to express the paternal relation, in the mixed Aramaean dialect of Palestine, during the New Testament age. Especially would it be naturally employed from infancy in addressing the male parent, like the modern papa; hence its occurrence in the New Testament only as a vocative (Winer, Gram. of the New-Test. Diction, § 29)'. Its reference to God (comp. Jeremiah 3:4; John 8:41) was common among the later Jews (Hamburger, Real-Encyklop. s.v.). To guard against the appearance of too great familiarity, however, the writers of the New Testament, instead of translating the title into its Greek equivalent, πάπα, have retained it in its foreign form — one of emphasis and dignity; but they have in all cases added its meaning, for the convenience of their merely Greek readers. Hence the phrase "Abba, father" in its two-fold form (Critica Biblica, 2:445).
2. Through faith in Christ all true Christians pass into the relation of sons; are permitted to address God with filial confidence in prayer; and to regard themselves as heirs of the heavenly inheritance. This adoption into the family of God inseparably follows our justification; and the power to call God our Father, in this special and appropriative sense, results from the inward testimony of our forgiveness given by the Holy Spirit. (See ADOPTION).
3. The word Abba in after ages came to be used in the Syriac, Coptic, and Ethiopic churches, in an improper sense, as a title given to their bishops (D'Herbelot, Bibl. Orient. s.v.), like padre, etc., in Roman Catholic countries.' The bishops themselves bestow the title Abba more eminently upon the Bishop of Alexandria; which gave occasion for the people to call him Baba, or Papa, that is, grandfather — a title which he bore before the Bishop of Rome.
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McClintock, John. Strong, James. Entry for 'Abba'. Cyclopedia of Biblical, Theological and Ecclesiastical Literature. https://www.studylight.org/encyclopedias/eng/tce/a/abba.html. Harper & Brothers. New York. 1870.