The Gospel of Matthew has a title ευαγγελιον κατα Μαθθαιον (“The Gospel according to Matthew”).
Martin Hengel, defends an early fixation of the Gospels’ manuscript titles in his book, The Four Gospels and the One Gospel of Jesus Christ.
More recently, Simon Gathercole, takes Hengel's argument one step further and argues it was an authentic reading in the earliest aspect of its textual transmission. He argues that while it is not in the main body, it is still part of the received text.
The hermeneutical implications of how to interpret this part of the text in Matthew is significant. For example, in an article on the Gospel of Mathew, Wikipedia notes:
The work does not identify its author, and the early tradition attributing it to the apostle Matthew is rejected by modern scholars.
Another article on Authorship of the Bible in relationship to the Gospel Matthew in Wikipedia notes:
The majority of modern scholars believe it is unlikely that this gospel was written by an eyewitness to the ministry of Jesus...
In relationship to the Historical Reliability of the Gospels, Wikipedia also notes:
Scholars doubt that the authors were the evangelists Matthew and Luke: it seems unlikely, for example, that Matthew would rely so heavily on Mark if its author had been an eyewitness to Jesus's ministry.
But if Gathercole is correct, the earliest text of Matthew contains the title, even though it is not in the body of the narrative. So, the hermeneutical question is relevant for New Testament studies.
The title, included in the text of Matthew, would work against the assumption that the Gospels resemble biographical literature such as the Life of Aesop and the Alexander Romance which are held to be anonymous.
Does the preposition κατα, mean “according to” in a strict limited sense or “handed down from” - i.e. in a broad sense of being passed down in the genre tradition of Matthew? If the former, would that indicate an explicit claim to authorship or at least editorial oversight in its final contents?
Gathercole notes how in early history genre it is more common for the main body to have the name of the author. However, he still adduces a number of examples showing this is not a rule. He also gives examples of biographies that don't mention their author in the main body: Josephus' Vita, Plutarch's biographies on Moses, etc.
So, from a form critical perspective, the hermeneutical understanding of the preposition κατα in its context is important.
Taking a strict view of the preposition meaning “according to” appears to support the view that the author himself put the title on his writing. After all, if the author of the Gospel was a tax collector than this might be an expected way to not write anonymously about Jesus.
Although, to be sure, the title could have been placed by a scribe after various notes were compiled and edited following Matthew’s death. Many of our modern books are published in a posthumous manner with the title of the author listed as if he or she were alive while writing the book.
Regarding the composition of Matthew, Eusebius (Historia Ecclesiastica 3.39.16) records Papias as stating:
But concerning Matthew he writes as follows: 'So then Matthew wrote the oracles in the Hebrew language, and every one interpreted them as he was able.'
ταῦτα μὲν οὖν ἱστόρηται τῷ Παπίᾳ περὶ τοῦ Μάρκου· περὶ δὲ τοῦ Ματθαῖου ταῦτ’ εἴρηται· Ματθαῖος μὲν οὖν Ἑβραΐδι διαλέκτῳ τὰ λόγια συνετάξατο, ἡρμήνευσεν δ’ αὐτὰ ὡς ἧν δυνατὸς ἕκαστος.
The understanding of the Papias use of "interpreted" is interesting. Anthony Thistleton presents detailed evidence concerning this use of διερμηνευω, even claiming that up to three quarters of the uses of διερμηνευωin by Philo means to "put into words." In other words it's a bit more than just meaning "to translate." It might also convey the idea of adding extra illustrations in from other sources to explain the oracles of Jesus. You can find Thistleton's thoughts on διερμηνευω expressed in an article of his in the Journal of Theological Studies, April 1979, pp. 13-36. Another discussion, related to the use of the word "interpretation" in Papias, can be found here.
In short, the proper contextual interpretation of the preposition κατα appears to me to be a useful hermeneutical question. That’s because some have argued that a late Greek version of Matthew was written by somebody besides Matthew. In other words, our Greek version could be an expanded interpretation (mentioned by Papias) of Mathew’s organized oracles of Jesus.