Bart Ehrman on his blog gives an interesting answer, based on the old scholarly consensus (at least of the majority of scholars) that Mark wrote the very first Gospel:
I think there may be one other thing going on with the NT Gospels that led their authors to write their accounts anonymously. I’ve never seen this suggested in the scholarly literature before, which either means I came up with it myself (in which case, caveat lector!) or I haven’t read enough scholarly literature. It is this. I think when Mark was writing his Gospel, he was imagining that he was continuing the story that he inherited from the Hebrew Bible. As you know, the final prophet of the Hebrew Bible, Malachi, ends by promising that Elijah would be coming before the “day of the Lord.” And how does Mark begin? By describing the coming of John the Baptist in the guise of Elijah. Mark is a continuation of the narrative of the Hebrew Bible.
But as you probably know, the Hebrew Bible – in the sequence of books given in the original Hebrew — does not end with Malachi, the final prophet, the way the English Old Testament does. It ends with 2 Chronicles, a narrative book that describes, at the very end, the fall of Jerusalem to the Babylonians and then the promise to rebuild the city by the Persian king Cyrus. There has been sin, and destruction, and the promise of restoration – told in a historical narrative. And Mark picks up the story at that point, with the coming then of the Savior, Jesus.
The historical books of the Hebrew Bible (Joshua, Judges, 1 and 2 Samuel, 1 and 2 Kings, 1 and 2 Chronicles) are anonymous. They are telling the history of the people of God, not based on the authority of the author but as a holy narrative of how God worked among his people. The names of the authors are unimportant and irrelevant in this kind of sacred history. Mark continues the sacred history, and like his predecessors, tells his story anonymously. Matthew and Luke and even John do it in their own ways, and also, as a result, tell their sacred history in the person of Jesus anonymously. I don’t think it’s surprising at all that they did not reveal their names. (read more here)
As you know, I do not subscribe to this scholarly consensus, but advocate that the first Gospel we know of is that of Marcion of Sinope - and in this light my answer differs from that of Bart Ehrman. In contrast to his and other scholars opinions, my own suggestion is based on an early Christian source, Tertullian of Carthage. In his refutation of Marcion, Tertullian complains about Marcion not having attached
‘to his Gospel’ an ‘author’s name’.
To Tertullian, publishing a text without also adding a title (titulum quoque affigere) was a form of dissimulation or pseudepigraphy. On Marcion’s explicit
criticism of those who had copied and added the names of Apostles and Apostolic
men to their rewriting of his text, Tertullian will come back in Adv. Marc. IV 5.5: ‘Marcion's complaint
is that the Apostles are held suspect of collusion and dissimulation, even to
the debasing of the Gospel’ (Apostolos praevaricationis et simulationis suspectos Marcion haberi queritur usque ad evangelii depravationem). Tertullian claims, Marcion’s Gospel,
contrary to what the reader expects, was a work without its ‘author’s name’ (debita auctoris), something that ‘gives
no promise of credibility’ as it misses ‘a fully descriptive title’. This argument,
however, only works if Tertullian had, indeed, assumed that the text was by
Marcion and, therefore, should have carried Marcion’s name. Tertullian did not buy into
Marcion’s explanation that ‘he might not assume permission to add a title
for it’. Why did Marcion withheld his name or why did he not pseudonymously add
the name of somebody else, Paul, for example, as Tertullian is going to suggest
in the next section (‘even if Marcion had introduced his Gospel under the name of Paul in person’)? As we can see from Marcion's collection of Pauline Letters which only contain letters which even modern
literary criticism attribute to Paul or the Pauline tradition, Marcion seems to
have been one of those authors and redactors who were keen on authenticity.
Perhaps, he did not add his own name to his Gospel
as he may have regarded this text as nothing else than the complementary
collection of narratives and sayings to go with his collection of Pauline Letters. And we may even
hypothesise that the Gospel might not
have gained the importance it got, if it had not been picked up by those ‘Apostles’
and ‘Apostolic men’ who, if Marcion is correct, nicked his text, copied,
altered and published it, just like him, without putting their names to their products. In this they may have only followed the Gospel they copied (and altered), namely that of Marcion. Nevertheless, people seem to have quickly added the names of apostles (Matthew, John) and apostolic men (Mark; Luke) to these texts, as, according to Tertullian, Marcion already complained, too, about such pseudonymous branding of these texts.
does not see any moral reason why the one who, according to Tertullian’s
account, had dared ‘to overturn the whole body’ shrank back in putting his own
name (or that of Paul) to this text. It is clear, however, from this argument,
that he therefore took Marcion as the Gospel’sauctor who’s name should have been
fixed to this text. This assumption is particularly made evident by Tertullian: ‘Marcion, on the other hand,
attaches to the Gospel, clearly his own, no author's name’. Had Tertullian intended to make a cynical
point, the argument made no sense. The only cynicism comes from the criticism
that Marcion who claims to be authentic and correct sees ‘no crime’ in turning
upside down (evertere) the entire
body of the Gospel.
If you wish to read more on this topic - you can now check my new monograph which has appeared a few days ago: