Abba has been found as a personal name in a first century burial at Giv'at ha-Mivtar and Abba also appears as a personal name frequently in the Gemara section of the Talmud, dating from 200-400. This would mean that Barabbas was the son of one named Abba.
Abba also means "father" in Aramaic. Jesus sometimes referred to God as "father;" Jesus' use of the Aramaic word Abba survives untranslated (in most English translations) in Mark 14:36. In the gospels, Jesus rarely refers to himself as "son of God" and never refers to himself as "son of the father.". However, some scholars like Michael Magee and Mary Whitehouse speculate that "bar-Abbâ" could refer to Jesus himself as "son of the father".
Hyam Maccoby and some other scholars have averred that Jesus was known as "bar-Abba", because of his custom of addressing God as 'Abba' in prayer, and referring to God as Abba in his preaching. It follows that when the Jewish crowd clamored before Pontius Pilate to "free Bar Abba" they could have meant Jesus. Anti-Semitic elements in the Christian church, the argument goes, altered the narrative to make it appear that the demand was for the freedom of somebody else (a brigand or insurrectionist) named "Barabbas". This was in, the theory goes, part of the tendency to shift the blame for the Crucifixion towards the Jews and away from the Romans. (See Hyam Maccoby, Revolution in Judea.)
Maccoby identifies Paul of Tarsus for this shifting of blame in The Mythmaker: Paul and the Invention Of Christianity, which explains extensively why it was necessary to appease Roman sentiment prior to Constantine I's Edict of Milan (Edict of Tolerance) in 313, which legalized Christianity.
The appeasing of Roman sentiment was, Maccoby suggests, confined to the matters of the blame for Crucifixion and over Jesus' "true" mission in life. Maccoby argues that Jesus was an anti-Roman revolutionary and that Paul, who had never met Jesus during his life-time, disagreed strongly with Jesus' actual followers over what Jesus' mission was.
In his role of Apostle to the Gentiles, Paul absolved Rome of any blame for the crucifixion so that new Roman converts could more easily accept the miracle of Jesus' resurrection with no guilt for the murder that made it possible. For a Roman convert to accept that Jesus was the messiah he would also be accepting that Rome killed God's only son - so Paul shifted the blame on to the Jews, and the Barabbas/Pilate story and, more famously, the Judas myth, were used as blame shifting tactics to get new recruits to Paul's newly formed religion.
Benjamin Urrutia, co-author with Guy Davenport of The Logia of Yeshua: The Sayings of Jesus agrees with Maccoby and others who aver that Yeshua Bar Abba or Jesus Barabbas must be none other than Jesus of Nazareth, and that the choice between two prisoners is a fiction. However, Urrutia opposes the notion that Jesus may have either led or planned a violent insurrection. Jesus was a strong advocate of "turning the other cheek" - which means not submission but strong and courageous, though nonviolent, defiance and resistance. Jesus, in this view, must have been the planner and leader of the Jewish nonviolent resistance to Pilate's plan to set up Roman Eagle standards on Jerusalem's Temple Mount. The story of this successful resistance is told by Josephus — who, curiously, does not say who was the leader, but does tell of Pilate's crucifixion of Jesus just two paragraphs later in a passage whose authenticity is heavily disputed. (See article Josephus on Jesus, in particular the section "Arabic Version." This version seems to be free of the postulated Christian interpolations, but still makes it clear that Pilate ordered the crucifixion of Jesus.)