1. I. Introduction (1-2)
Thematic Relation: In recognizing Gaius' spiritual condition, and praying for his health, John encourages Gaius to persevere in well-doing.

Some Key Words

Gaius (gaios [1050]):
Lord, see notes above; a Christian
Love (agapao [25]):
to direct one's will toward and find one's joy in a thing, love that expresses compassion; to be fond of, to welcome; to love in a moral sense
Truth (aletheia [225]):
the unveiled reality agreeing with the appearance, the essence, integrity; what is true in any matter, moral truth, personal candor, free of pretence and deceit; truth
Beloved (agapetos [27]):
dear, an object of joy and compassionate love; esteemed, favorite, worthy of love
Prosper (euodoo [2137]):
to lead by a direct and easy way, to grant success, to be successful; to help on the road, succeed in reaching, succeed in business affairs
Be in Good Health (hugiaino [5198]):
to be healthy, sound, well; to hold opinions that are free from error; to be well in body, uncorrupt in doctrine
Soul (psuche [5590]):
the immaterial part of life common to man and animal, as distinct from spirit; breath of life, the seat of feelings, that which designates a moral being designed for eternal life, the part which is not dissolved by death; sentience, not yet the immortal soul, but still more than mere animal vitality
 

Paraphrase

1 From the elder to my good friend Gaius, truly beloved. 2 I pray that your life be as blessed in all respects as I know it to be spiritually.

Key Verse

2 - The prayer is for full blessedness in every aspect of life, physical and eternal.

Thematic Relevance:

 To know our spiritual growth is recognized (and not just our own heart lying to us) encourages us. To know we have the prayers of others strengthens us to continue.

Doctrinal Relevance:

 We ought to pray for each other.

Moral Relevance:

 When we are true to God, when we are walking out our faith, these things are known to those who know God. We don't need to advertise it. Godly actions advertise themselves sufficiently.

People Mentioned:
Gaius
There is a Gaius mentioned as having come from Macedonia to Ephesus with Paul (Ac 19:29), where he was dragged off before the magistrate. He is noted as being from Derbe (Ac 20:4), which is east of Ephesus. Gaius is also noted in Ro 16:23 as hosting the church in the place where Erastus was city treasurer. There's another Erastus associated with Timothy, but I don't think that would necessarily be the same one. There is also a Gaius noted in 1Co 1:14. If Erastus the treasurer is also Erastus the traveler, then that would equate these last two references to Gaius. Looking at the maps of Paul's trips, it would be hard to make all of these references match a single Gaius. Two, I could see. That would give one who was from Derbe, and had accompanied Paul to Macedonia. Trying to make any one of these the one to whom John writes seems a bit difficult, and would be no more than conjecture.

Some Parallel Verses

1
2Jn 1 - The elder writes to the chosen lady. Ac 19:29 - There was a Gaius accosted in Ephesus, Ac 20:4 - a Gaius who was from Derbe, Ro 16:23 - a Gaius who hosted Paul, and 1Co 1:14 - one baptized by Paul. 1Jn 3:18 - Love must not consist of mere words, but must be true and active. 2Jn 1 - The lady was also loved in such truth.
2
 

New Thoughts

Hmm. An interesting connection here. John - who focuses so much on the need for love to be active if it be true - professes a true love for Gaius (as he did to the lady, as well). It occurs to me that both of these letters were seen by him as acts of that love. Both sought to encourage, edify, and warn the recipient. When we see a brother doing well, active love will encourage and commend him. When we see him surrounded by dangers, active love will warn him of what is there just in case he's not aware of it. Active love will show him the way to steer clear of those dangers. God, help my love for my family to be such an active love.