In short, no, the evidence suggests that in the first century AD and beyond, it just meant “father”.
Here are the three uses of the term in the New Testament:
For you did not receive a spirit of slavery to fall back into fear, but you have received a spirit of adoption. When we cry, “Abba! Father!” Romans 8:15
And because you are children, God has sent the Spirit of his Son into our hearts, crying, “Abba! Father!” Galatians 4:6
[At Gethsemane, Jesus said], “Abba, Father, for you all things are possible; remove this cup from me; yet, not what I want, but what you want.”
Linguistic Background: In the New Testament, it is represented by the transliterated as αββα/abba, that is, Greek letters that represent Aramaic sounds. Abba (אַבָּא) is a variation of Ab (אַב), “padre”. Abba is not a Hebrew word, but an Aramaic one, the language of gentiles and Jews in the western Mediterranean, and also of the rabbis through many centuries. It started out as a diminutive form of Ab and originally meant “papa”; nevertheless, by the process of evolution that all words undergo, it came to be the form that both children and adults used to address their fathers. Part of that process was that, the Jews began to use Abba as a title for outstanding rabbis, for example, Abba Saul in the second century AD. Those rabbis were “fathers”, not “papas”! And for that reason Jesus taught: “But you are not to be called rabbi, for you have one teacher, and you are all students. And call no one your father on earth, for you have one Father—the one in heaven.” (Matthew 23:8-9). Matthew uses the Greek term, but Jesus must have been referring to their usage of Abba. There is, by the way, not a single reference to non-Christian Jews using Abba in prayer.
Use by Jesus: The evidence indicates that Jesus always taught in Aramaic, a Semitic dialect that is a cousin to Hebrew and the forerunner of the 2nd century AD Syriac dialect of the Peshitta. Some examples where the evangelists quote him directly speaking in Aramaic: Talitha cumi (Mark 5:41);  Ephphatha (Mark 7:34). He also used Aramaic as at least part of his language for prayer, as shown by his calling the Father Abba (Mark 14:36). This was the language of many early Christians, and some words passed directly to Greek: besides Abba, there is Marana tha (“Our Lord, come!”, 1 Cor 16:22). On the other hand, after the New Testament the term Abba is not found in the church fathers, except in quotes of the New Testament references.  In Romans and Galatians, Paul goes ahead and translates Abba from the Aramaic to Greek, and he uses the normal word for father, patēr/πατήρ. That is, he translated it “father”, not “daddy” or “papa”. Hence in Rom 8:15, αββα ὁ πατήρ. The Lord’s Prayer also uses patēr, Father.
Current Usage: It has become the custom in English, but also very popularly in Latin America, and I have found references in French and German, to translate Abba as Papa.
The best evidence says that Abba simply means “Father”. But the gospel comforts us that that is more than enough truth that “we are bold to say, Our Father”.
* This information is taken from my Romans commentary in the Comentario Bíblico Contemporáneo.
 Talitha too is a diminutive form which Mark translates into Greek as “little girl”. It was probably means to be affectionate, since a 12-year-old was not considered “little”. Was he saying, “Get up, sweetheart”?
 Here is a rich reference from Irenaeus, Against heresies 5.8.1 [ANF 1:553] – “For those to whom [Paul in Romans 8] was writing were not without flesh, but they were those who had received the Spirit of God, ‘by which we cry, Abba, Father.’ If therefore, at the present time, having the earnest, we do cry, ‘Abba, Father,’ what shall it be when, on rising again, we behold Him face to face; when all the members shall burst out into a continuous hymn of triumph, glorifying Him who raised them from the dead, and gave the gift of eternal life? For if the earnest, gathering man into itself, does even now cause him to cry, ‘Abba, Father,’ what shall the complete grace of the Spirit effect, which shall be given to men by God? It will render us like unto Him, and accomplish the will of the Father; for it shall make man after the image and likeness of God.”
“Does ‘Abba’ mean ‘Daddy’?” by Gary S. Shogren, PhD in New Testament Exegesis, Profesor of Seminario ESEPA, San José, Costa Rica