So, for those of you who aren't familiar with this particular meme, it is common to find around the Internet and in sermons throughout the world that where Jesus is recorded in the New Testament to use the Aramaic word "abba" that the term was an informal word, the likes a child would refer to their pop (i.e. "dad" or "daddy").
This stemmed from an idea that was originally proposed by a scholar named Joachim Jeremias (b1900-d1979); mainly, that the form "abba" originated from "child-babble." The connection between "abba" and "daddy" was then popularized by his following.
However, this idea was immediately challenged by a number of other scholars, such as James Barr who published an article entitled "Abba Isn't 'Daddy'" (in the Journal of Theological Studies) which outlined the numerous problems with such an assertion and addressed them in detail.
Overall, I believe that Mary Rose D'Angelo summed up what happened next nicely:
"Jeremias began almost at once to retreat from the claim that "abba" had the same connotations as "daddy." In a sense, Barr's title (but only his title) misrepresents Jeremias. Even as Jeremias acknowledged that the word was in common use by adults and was used as a mark of repect for old men and for teachers, he continued to stress the origins in babytalk and the consequent intimacy as a special component of Jesus' use of the word. This meaning seems to have been the basis on which he regarded Jesus' use as absolutely distinct from the Judaism of his time.
The NT itself gives quite a different reading of αββα. Each of the three occurrences of αββα in the NT is followed by the Greek translation ο πατερ, "the father." This translation makes clear its meaning to the writers; the form is a literal translation -- "father" plus a definite article -- and like abba can also be a vocative. But it is not a diminutive of "babytalk" form. There are Greek diminutives of father (e.g., παππας [pappas]), and the community chose not to use them.
--Mary Rose D'Angelo. Journal of Biblical Literature, Vol. 111, No. 4 (Winter, 1992), pp. 615-616
And beyond this, many years after Jeremias' death, modern linguistic study of how children pick up speech has completely discounted his conclusions of abba as "babytalk."
There is still a point of confusion: In Modern Hebrew, "abba" has become commonly used as... You guessed it: "Daddy." So, when a Hebrew speaker happens upon this anecdote, to them it makes "perfect sense." :-)
The myth survives.
Nevertheless, happy Father's Day to all of the pas, fathers, pops, dads, papas and daddies out there, wherever you are!