Abba means "my father" and not "daddy' or "dad" or "papa"

Here is evidence Abba does not mean "daddy" we see Joseph addressing Pharoah talking about his father and brothers, and the accompanying Aramaic translation.  The Hebrew has avi  , the Aramaic has aba אַבָּא 

 

Then Joseph went in and told Pharaoh, and said: 'My father and my brethren, and their flocks, and their herds, and all that they have, are come out of the land of Canaan; and, behold, they are in the land of Goshen.' --Genesis 47:1

 

וַיָּבֹא יוֹסֵף, וַיַּגֵּד לְפַרְעֹה, וַיֹּאמֶר אָבִי וְאַחַי וְצֹאנָם וּבְקָרָם וְכָל-אֲשֶׁר לָהֶם, בָּאוּ מֵאֶרֶץ כְּנָעַן; וְהִנָּם, בְּאֶרֶץ גֹּשֶׁן--Genesis 47:1 Masoretic Hebrew text
 

וַאֲתָא יוֹסֵף, וְחַוִּי לְפַרְעֹה, וַאֲמַר אַבָּא וְאַחַי וְעָנְהוֹן וְתוֹרֵיהוֹן וְכָל דִּילְהוֹן, אֲתוֹ מֵאַרְעָא דִּכְנָעַן; וְהָא אִנּוּן, בְּאַרְעָא דְּגֹשֶׁן. --Genesis 47:1 Onkelos Aramaic Translation using Hebrew font

The Hebrew text reads Avi אָבִי  , 'my father', the Aramaic reads אַבָּא aba, though another site has it as אַבָה --ava.
 

Now if Abba/Aba/a means "daddy" how silly would it be for Joseph--the prime minister of Egypt to address his boss concerning his father and brother using the word "daddy"? It would have been embarrassing! Granted Joseph probably did not use either Abba or Avi, since he would have been speaking Coptic to the Pharoah, however the Hebrew bible and the Aramaic translation of it use the ORDINARY word for "my father" which are avi and aba/ava.

 

Furthermore, Barabbas, has the word abba in it, yet no one supposes this means "son of daddy,"

ἦν δὲ ὁ λεγόμενος Βαραββᾶς μετὰ τῶν στασιαστῶν δεδεμένος οἵτινες ἐν τῇ στάσει φόνον πεποιήκεισαν.--Mark 15:7 Greek (almost every manuscript)

The man named Barabbas had been imprisoned with the insurrectionists who had committed murder in the insurrection.--Mark 15:7 NASB

 ܘܐܣܝܪ ܗܘܐ ܓܒܪܐ ܥܒܕ ܒܝܫܐ ܕܡܬܩܪܐ ܒܪ ܐܒܐ ܘܐܝܬ ܗܘܐ ܓܒܪܐ ܕܥܒܕ ܒܝܫܬܐ ܘܩܛܠ ܩܛܠܐ--Mark 15:7, Old Syriac Sinaiticus

 ܘܐܝܬܼ ܗ̱ܘܐ ܚܕܼ ܕܿܡܬܼܩܪܐ܂ ܒܿܪ ܐܒܿܐ܂ ܕܿܐܣܝܪ ܗ̱ܘܐ ܥܡ ܥܒܼܕܿܝ̈ ܐܣܛܣܝܢ܂ ܗܢܘܼܢ ܕܿܩܛܠܐ ܒܿܐܣܛܣܝܢ ܥܒܼܕܼܘ--Mark 15:7 Peshitta Text

 We see the Aramaic texts (old Syriac and Peshitta) have Aba/Ava spelled the same as the Aramaic text of Genesis 47:1 "וַאֲתָא יוֹסֵף, וְחַוִּי לְפַרְעֹה, וַאֲמַר אַבָּא וְאַחַי " Here is how two popular Protestant lexicons explain the name "Bar Abbas"

Thayer

Barabbas = "son of a father or master"
1) the captive robber whom the Jews begged Pilate to release instead of Christ
Part of Speech: noun proper masculine

Strong

G912
Of Chaldee origin ([H1347] and G5 (Greek)); son of Abba; Bar-abbas, an Israelite: - Barabbas.

We see it is either a name, or just how to say "father", and if it is a name, its not likely a parent will name their son "daddy" (unless they're Hispanic, but even it would be a petname).
 

Now, let's go to the texts where the word "Abba" appears in Greek and English untranslated: 

And He was saying, “Abba! Father! All things are possible for You; remove this cup from Me; yet not what I will, but what You will.”--Mark 14:36 NASB

 

The Aramaic texts of this gospel read:

ܘܸܐܡܲܪ܂ ܐܲܒܼܵܐ   ܐܵܒܼܝ܃   ܟܿܽܠ ܡܸܕܸܿܡ ܡܸܫܟܿܲܚ ܐܲܢ̱ܬܿ܂ ܐܲܥܒܿܲܪ ܡܸܢܝ ܟܿܵܣܵܐ ܗܵܢܵܐ܂ ܐܸܠܵܐ ܠܵܐ ܨܸܒܼܝܵܢܝ ܕܼܿܝܠܝ܂ ܐܸܠܵܐ ܕܼܿܝܠܵܟܼ--Mark 14:36 Peshitta Text

And he said, Father, my Father, thou canst (do) every thing, cause to pass from me this cup: but not my will, but thine.--Mark 14:36, Dr. John W. Etheridge's English Peshitta translation  

ܘܐܡܪ ܐܒܝ ܟܘܠ ܡܕܡ ܒܐ̈ܝܕܝܟ ܡܛܐ ܐܥܒܪ ܡܢܝ ܟܣܐ ܗܢܐ ܐܠܐ ܠܐ ܨܒܝܢܝ ܕܝܠܝ ܢܗܘܐ ܐܠܐ ܕܝܠܟ--Mark 14:36 old Syriac Sinaiticus 

 "And He said, My Father, everything is in your hands, it has arrived. Make this cup pass from me, yet not according to my will, but according to your will."--Mark 14:36 translation of Old Syriac Sinaiticus page 364

 Elsewhere in the Peshitta text of Mark we see the SAME EXACT spelling of Aba being used

 ܥܢܵܐ ܝܸܫܘܼܥ܂ ܘܸܐܡܲܪ܂ ܐܲܡܼܝܢ ܐܵܡܲܪ ܐ̱ܢܵܐ ܠܟܼܘܼܢ܃ ܕܿܠܲܝܬܿ ܐ̱ܢܵܫ ܕܿܫܵܒܼܸܩ ܒܿܵܬܸܿܐ̈ ܐܲܘ ܐܲܚܸܐ̈ ܐܲܘ ܐܲܚܘܵܬܼܵܐ̈܃ ܐܲܘ   ܐܲܒܼܵܐ   ܐܲܘ ܐܸܡܵܐ܃ ܐܲܘ ܐܲܢ̱ܬܿܬܼܵܐ ܐܲܘ ܒܿܢܲܝܵܐ̈܃ ܐܲܘ ܩܘܼܪܝܵܐ̈܂ ܡܸܛܽܠܵܬܼܝ܂ ܘܡܸܛܽܠ ܣܒܼܲܪܬܼܝ--Mark 10:29

 Jeshu responded and said, Amen I say to you, There is no man who forsaketh houses, or brethren, or sisters, or father, or mother, or wife, or children, or fields, on account of me, and on account of my gospel, --Mark 10:29 Dr. John W. Etheridge's English Peshitta translation

The Old Syriac has verses with the same spelling:

 ܢܫܠܡ ܓܝܪ ܐܚܐ ܠܐܚܘܗܝ ܠܡܘܬܐ   ܘܐܒܐ   ܠܒܪܗ ܘܢܩܘܡܘܢ ܒܢ̈ܝܐ ܥܠ ܐܒܗܬܐ ܘܢܡܝܬܘܢ ܐܢܘܢ--Mark 13:12 Old Syriac Sinaiticus

"For a brother will betray his brother to death, and a father [ܘܐܒܐ] his son, and children will rise up against their parents and kill them."--Mark 13:12 translation from The Old Syriac Gospels: Matthew and Mark

 Now for reference the other times Abba appears in the Greek texts of the New Testament untranslated:

 οὐ γὰρ ἐλάβετε πνεῦμα δουλείας πάλιν εἰς φόβον, ἀλλὰ ἐλάβετε πνεῦμα υἱοθεσίας ἐν ᾧ κράζομεν· Αββα ὁ πατήρ·--Romans 8:15

For you have not received a spirit of slavery leading to fear again, but you have received a spirit of adoption as sons by which we cry out, “Abba! Father!”--Romans 8:15 NASB

ὅτι δέ ἐστε υἱοί, ἐξαπέστειλεν ὁ θεὸς τὸ πνεῦμα τοῦ υἱοῦ αὐτοῦ εἰς τὰς καρδίας ἡμῶν, κρᾶζον· Αββα ὁ πατήρ.--Galatians 4:6  

Because you are sons, God has sent forth the Spirit of His Son into our hearts, crying, “Abba! Father!”--Galatians 4:6 NASB

 Finally, let's looks how dictionaries, lexicons, concordances refer to it as:

Thayer:

Abba = "father"
1) father, customary title used of God in prayer. Whenever it occurs in the New Testament it has the Greek interpretation joined to it, that is apparently to be explained by the fact that the Chaldee "ABBA" through frequent use in prayer, gradually acquired the nature of a most sacred proper name, to which the Greek speaking Jews added the name from their own tongue.
Part of Speech: noun
Citing in TDNT: 1:5, 1

Strong:

G5
Of Chaldee origin [H2]; father (as a vocative): - Abba. 

Louw-Nida:

Gloss	Section
Father (a title for God)	12.12

From Hebrew Union College: 

 1 father Com. --(a) of humans Com. (a.1) honored senior, chief, master OfA-Egypt, JLAtg. (a.2) chief monk Syr. --(b) Heavenly, i.e. God (see also below s.v. ˀabbā) JLAtg, PTA, CPA, Syr. --(c) of animals: sire JBA.
  2 ancestor, forefather Com. --(a) pl. parents Qum, Syr, JBA. --(b) in particular, the three forefathers Abraham, Isaac, and Jacob (also sg. for any one of them) Qum, JLAtg, Gal, PTA, JBA. --(c) progenitor, spiritual founder PTA. --(d) the fathers of the Church Syr.
  3 ˀabbā : my father! (vocative) JLAtg, Gal, PTA, CPA, Syr, JBAmb.
  4 fig. Qum, JBA. --(a) creator, source Qum. --(b) common factor JBA. --(c) 'source' as an architectural term JLAtg.
  5 byt ˀb, ܒܝܬ ܐܒ : family, clan OA, Qum, Syr.
  6 as a personal name Syr, JBA.
  7 in compounds with relational terms . --(a) ˀb ˀbwhy: grandfather or ancestor Palm. --(b) אב דאב : paternal grandfather PTA, JBA. --(c) אבא דאימא : maternal grandfather JBA. --(d) אחא דאבא٠ אח אב : paternal uncle : see s.v. ˀḥb JBA.

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Notice all of these say its just a way of saying father, biological or not and make no mention of it being a way of how a small child says "daddy." Yet sites like "gotquestions.com" ignorantly spread this "fact" around.

The blame and origin of claim "abba" means daddy seems to generally be placed on German Lutheran scholar Joachim Jeremias who gave this impression.