What Richard Carrier has done that no mythicist before him was able to do is show a clear path for the origin of a Mythical Jesus. It's not that Carrier found new material, but that the experts who dealt with it before him did not (as far as I can see) notice the mythicist potential.
Here's what I think is his best argument, in a nutshell.
Philo of Alexandria (about 25 BCE – c. 50 CE) was an Alexandrian Jew, and the foremost Greco-Jewish philosopher. His life's work was syncretizing Greek and Jewish thought.
Philo follows the usual Platonic system with an Absolute (G-d) and a Demiurge but he shifts it just a bit. Instead of having the Demiurge as creator, he has the Demiurge (which he calls the Logos) as the agent of creation.
"For nothing mortal can be made in the likeness of the Most High One and Father of the Universe, but only in that of the second God who is his Logos" (Philo, Questions on Genesis, 11.62)
For Philo, the High Priest on earth is a reflection of the Logos.
"For there are, as is evident, two temples of God: one of them this universe in which there is also as high priest his Firstborn, the Divine Logos, and the other the rational soul, whose priest is the real man" (On Dreams, 1.215).
Then there is a passage in Zechariah where he has a vision that gives him G-d's instructions for the restoring the the Temple and its priesthood.
"Take the silver and gold [from the exiles] and make a crown, and set it on the head of the high priest, Joshua son of Jozadak (Yeshua son of Yehozadak). Tell him this is what the Lord Almighty says: ‘Here is the man whose name is the Branch, and he will branch out from his place and build the temple of the Lord. It is he who will build the temple of the Lord, and he will be clothed with majesty and will sit and rule on his throne." (Zechariah 6:11-13)
Philo's handling of this passage is a bit problematic because of differences between the Hebrew text and the Greek translation he is using, but he seems to find a Messianic message in it, or at least a proof about the nature of the universe.
"I have also heard of one of the companions of Moses having uttered such a speech as this: "Behold, a man whose name is the East!" (Zechariah 6:12.} A very novel appellation indeed, if you consider it as spoken of a man who is compounded of body and soul; but if you look upon it as applied to that incorporeal being who in no respect differs from the divine image, you will then agree that the name of the east has been given to him with great felicity. For the Father of the universe has caused him to spring up as the eldest son, whom, in another passage, he calls the firstborn; and he who is thus born, imitating the ways of his father, has formed such and such species, looking to his archetypal patterns." (Philo of Alexandria, Confusion of Tongues, 14.62-63).
In other words, Philo seems to be saying that this man whose name is East (or Rising or Branch, depending on the translation) in Zechariah 6:12, is the same person as the Logos and (perhaps) the High Priest Joshua and his associates are "men symbolic of things to come" and when G-d brings his servant, the Branch, he "will remove the sin of this land in a single day." (Zechariah 3:8).
It takes a bit of close reading to get the point here. Carrier believes that Philo reinterpreted this passage to make the High Priest Joshua the same as the Branch. That seems plausible considering Philo's idea that the High Priest is the earthly reflection of the Logos, but I don't think it's a necessary part of the argument.
What I find most interesting about this type of analysis is that it breaks out of the box of textual criticism. Philo's dates overlap both Jesus and Paul. It is very plausible that Paul, working as an agent for the Saducees (Hellenized Jews) who controlled the Temple, was himself a Hellenized Jew and perhaps a follower of Philo's system.
And here we have Philo talking about a Logos, or second G-d, for whom the high priest is a type, and apparently with Messianic overtones. Moreover, it's a straight short, although maybe a bit speculative, to link this Logos with the name Jesus (Joshua).