Gaius was the Greek way of writing the Roman name Caius. Believe it or not, there are many references to Caius in Scripture.
Here in 3 John, we have yet another Caius who is probably different from all of the above! I never realized how popular this name was in the Bible. The Caius mentioned here by John has high standing just as many of these others who shared his name.
John counts him among his children. This could have signified many things during that time. Obviously, it would have meant they were close. However, it also could have meant he was a convert of John and was discipled by him. In that time, it was not uncommon for the elder to consider the younger a child by their authority in the faith. John would have been an aged man at the time of this letter.
Overall, this letter serves as an encouragement for some and a rebuke for others. Like other letters, the purpose was to kindle the fire that was authentic and call out the false teaching that was trying to infiltrate the church.