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.

 

Each Shall die for his own sin

Parents are not to be put to death for their children, nor children put to death for their parents; each will die for their own sin. -Deuteronomy 24:16

The idea that Christ died as if He commited our sin violates the Biblical teaching that each man be put to death for his OWN sins.  This verse, however, seems to only apply to adults, since young children in the Bible are put to death, or killed, because of their parents sins, as in the case of David's son, or the nations God commanded the Israelites to destroys.  Regardless, Christ could not have been put to death for the sins of others (at least justly!). Rather, Christ willingly laid down His own life for the sake of others, not that He should die as their substitute, but that by His righteous death, God would show mercy to mankind, thus allowing man forgiveness on the condition they repent of their sins.  After all, Christ did make several warning statements that certain sins would cause the soul to go to Hell forever, as we see in Matthew 5 with sexual sins, Matthew 6 with withholding forgiving your brother's sins and so on.

On another point, Deuteronomy 24:16 is a verse anti/counter-Missionary Jews use against Christians.  Their claim is legitimate, but only against baptistic and Calvinistic types of Christians that espouse this particular view of the atonement that says "Christ died in our place" or "He took the wrath of God in our place" and so on.  Rather, the Biblical Christ was a propitiation for the sins of the world, meaning His death provided MERCY for mankind. 

I have already written before that the idea of a penal substitution when logically followed out is a denial of Trinitarianism, or can be an affirmation of a sort of Arianism (since God cannot be forsaken by God), or some form of theology where Christ as a man and Christ as God are divorced (undoing the incarnation).

Calvinists try to prove a penaly substitution with verses like 2 Corinthians 5:21, or Isaiah 53 and so on.  When the fact is 2 Corinthians 5:21 by saying Christ "became sin for us" is an expression St Paul was using for a sin offering, and Isaiah 53 doesn't mean God treated Christ as if He had our sins, in fact Matthew 8:17 interprets this part of Isaiah as Jesus healing the sick and casting out demons:

When it was evening, they brought him many who were possessed by demons, and he drove out the spirits by a word and cured all the sick, to fulfill what had been said by Isaiah the prophet: “He took away our infirmities and bore our diseases.”--Matthew 8:16-17

Call no man father?

"Call no one on earth your father; you have but one Father in heaven."—Matthew 23:9

This verse has been used by anticatholics to claim it to be wrong to call a priest (or anyone) father, the verse previous would logically also prohibit calling men "doctor" since it means "teacher," but if Jesus was prohibiting calling men father then what about.....

"And say, if we had been in the days of OUR FATHERS..." (Matthew 23:30)

"...so worship I the God of my FATHERS..." (Acts 24:14)

When Elisha saw it happen he cried out, "My father! my father!* Israel's chariots and drivers!" But when he could no longer see him, Elisha gripped his own garment and tore it in two.(2 kings 2:12) *traditionally Jews regard both Elijah and Elisha as celibates.

When the king of Israel saw them, he asked, "Shall I kill them, my father?" (2 kings 6:21)

Hazael went to visit him, carrying a present, and with forty camel loads of the best goods of Damascus. On his arrival, he stood before the prophet and said, "Your son Ben-hadad, king of Aram, has sent me to ask you whether he will recover from his sickness."(2 kings 8:9)

"Stay with me," Micah said to him. "Be father and priest to me, and I will give you ten silver shekels a year, a set of garments, and your food."(judges 17:10)

"But Timothy's worth you know, how as a son with a father he has served with me in the gospel." (Philippians 2:22)

"Look to Abraham, your father..." (Isaiah 51:2) **Abraham MEANS "father of many nations"**

And he replied, "My brothers and fathers, listen. The God of glory appeared to our father Abraham while he was in Mesopotamia, before he had settled in Haran, (Acts 7:2)

“Thus he was to be the father of all the uncircumcised who believe”—Romans 4:11 *Thos is an OBVIOUS reference to Abraham as being a spiritual father

"as well as the father of the circumcised who not only are circumcised, but also follow the path of faith that our father Abraham walked while still uncircumcised." (Romans 4:12)

"For this reason, it depends on faith, so that it may be a gift, and the promise may be guaranteed to all his descendants, not to those who only adhere to the law but to those who follow the faith of Abraham, who is the father of all of us," (Romans 4:16)

"Was not Abraham our father justified by works when he offered his son Isaac upon the altar?" (James 2:21)

"To Timothy, my true child in the faith: Grace, mercy, and peace from God the Father and Christ Jesus our Lord." (1 Tim. 1:2)

"Do not rebuke an older man, but appeal to him as a father. Treat younger men as brothers" (1 Tim. 5:1)

"To Timothy, my beloved child: Grace, mercy, and peace from God the Father and Christ Jesus our Lord." (2 Tim. 1:2

"To Titus, my true child in a common faith: Grace and peace from God the Father and Christ Jesus our Savior." (Tit. 1:4)

I am writing to you, fathers, because you know him who is from the beginning. I am writing to you, young men, because you have conquered the evil one. (1 John 2:13)

I do not write this to make you ashamed, but to admonish you as my beloved children. For though you have countless guides in Christ, you do not have manyfathers. For I became your father in Christ Jesus through the gospel. (1 Cor. 4:14-15) *Literally reads “I begat you in Christ..”

"So God created man in his own image, in the image of God he created him... then the LORD God formed man of dust from the ground, and breathed into his nostrils the breath of life; and man became a living being." (Gen. 1:27, 2:7)

"When Adam had lived a hundred and thirty years, he became the father of a son in his own likeness, after his image, and named him Seth." (Gen. 5:3)

"Jesus said to them again, 'Peace be with you. As the Father has sent me, even so I send you.' And when he had said this, he breathed on them, and said to them, 'Receive the Holy Spirit.'" (John 20:21-22)

"Hence I remind you to rekindle the gift of God that is within you through the laying on of my hands... guard the truth that has been entrusted to you by the Holy Spirit who dwells within us." (2 Tim. 1:6, 14)

Neglect not the grace that is in thee, which was given thee by prophesy, with imposition of the hands of the priesthood. (1 Tim 4:14)

Conclusion: Call no man father was not a prohibition against the use of the word father to human beings, if it were well....everyone of the Apostles and God Himself would be guilty of this.