The earliest tradition could have been any eye witness To the works of Jesus. Yet as a specific tradition passed down from the non-disciple. I have not read about in any of the apocrypha. However the tradition is explained in both Matthew and Luke (Matthew 5:39). Learning how to cast out a demon, may help you gain perspective on how this non-disciple did this.
First we must really understand what a demon / evil spirit / unclean spirit means. Think about the word possession. In this context the word possess means "To have as a quality, characteristic" [a]. Next one must really understand what your spirit means. In the context of what the New Testament uses the word spirit the definition is "A mood or an emotional state" [b]. For example the three parts of God, his Mind, his Word / Flesh, his mood or an emotional state / his Love.
We all are made in his image. So we also have these three parts of us as well. When a thought is in our mind, we have a choice. This choice is whether or not we will speak that thought. For the word only does what the father tells him. Our mood is a reflection to our thoughts what is being said and what is being heard (1 Corinthians 2:11).
So then what does it mean to be of an evil spirit, the simple answer is a "bad mood". Now if one is in a bad mood, they might say things they don't truly mean these things said are Lies. Lies as we know are evil. Therefore a solid connection between a bad mood and an evil mood can be established. A bad mood without sin would logically be an unclean spirit.
Then what is a demon? In standard Christianity, a demon is defined as a fallen angel. But the Greek definition for the word translated as demon means "to be under the power of a demon." So when you see the translation a "demon possessed man". Another way that could be translated would be "A man that has a quality or characteristic of being under the power of a demon." Or simply someone doing what the devil wants and not what God wants. If that trait can be found on you, then you are in possession of that trait. That trait being evil, doing as that trait desires shows who has the control. If a demon can be identified by its traits of doing evil, and you have that trait as well, then the identity of the demon controlling you can be identified. Yet the control or power is given in free will. So blame can still fall on the man.
But since we are much more accustomed today with the word mood. To learn how to expel demons it will be much easier to explain using the word mood. So recorded in Luke is a man who also cast away someones bad mood. He did this using the name of Jesus. Much confusion on using the name of Jesus is active today. If you asked most people how to use the name of Jesus, you would most likely get "do this in the name of Jesus!" or "In the name of Jesus I command you!"
So next, to really understand how this "non-follower" used Jesus name. We go back in time to the time of Jesus. Doing something in the "old Jewish form of someones name" in today's standards is similar to a job position title. For example we expect a certain persona when we hear the word Waitress. We can tell that there is a lot of difference we we think of a Politician, or a Used Car Sales Man.
Well in those days they did not have a copyright office so specific talents where associated to ones name. Hopefully this helps you understand why the Disciples tried to stop the man because he wasn't one of them (Luke 9:49). Back in those days if you tried to use someones name, it would be similar to copyright infringement. The Disciples truly felt that the man was stealing away Jesus identity, by treating the guy in a bad mood the way Jesus did. That also is why they felt it was important to tell Jesus what they had done. So as to make him aware that his identity was at stake.
The use of someones name would have specific characteristics, for example posture, methods of doing things, fancy sayings, ect. We still have a bit of that today. If I lower my voice and say "I'll be back" you can see that I was impersonating Arnold Schwarzenegger. Or in Jewish days I would have been in the name of Arnold. Luck for us we are all called to impersonate Jesus. By doing this we can be saved.
But he has told us his name, bless those that curse you, turn the other cheek, listen, and love. This is how he got rid of those bad moods. So this non-disciple was pretty smart, he was able to understand the advantages of what Jesus was doing. The question of the traditions should be in the light now. Let us all go and cast out some demons =)