It literally says that Timothy is a genuine child, or genuine son in faith. In the King James my is italicized, because my is not there in the Greek.

We see a father son relationship between Paul and Timothy in Phil 2:22, which came before 1 Timothy, which may have affected the translators. Here, though, I think he is emphasizing that Timothy is genuinely a son, or a child, of God.

Can we ever know for sure that someone is saved? Paul knew for sure that Timothy was. We can't always know, though, as the end of chapter 5 shows us. We have to be very careful to know that our confidence in someone is coming from the Lord, and not from our feelings. Many who seem to be saved are not. Likewise, we don't want to hurt a real Christian by writing them off to quickly.

Paul desired God's best for Timothy. In his absence he wanted to communicate this to Timothy right away. This was more than mere formalities. Yes, letters of that era did have introductions similar to this, but Paul was moved by the Holy Spirit. Many people ask "how you doing" and then hardly even listen to the answer. We can ask people how they are doing because we are moved by the Holy Spirit, and we want to show them the love of God.

He wants Timothy to have the following things: (and he knows Timothy can have them because he is a true child of faith. We obtain these things by faith)

Grace: Undeserved goodness given to us, as a gift. We don't deserve it. By faith we have access to good things we didn't earn. Paul is caught up in wanting Timothy to have good things. Truthfully, we often don't care very much if people have extra goodness. God, please help us want other people's best.

We also need grace to help us obey God. (Heb 4:16) Grace helps us obey God. In this context I like the word "help" to describe grace.

Mercy: Holding back the bad we deserve. We should be repaid evil for evil from God. We need mercy every time we offend God. Paul was caught up in Timothy not receiving the punishment he deserved. It's good to not want people to receive the bad they deserve, because we will be shown mercy by the mercy that we show. We often want people to be punished, when we instead ought to apply our efforts to help them obtain God's mercy.

Peace: Not being at war anymore with God. When we sin, we need to receive forgiveness in order to be at peace with God. God doesn't want us to have any lingering, unresolved problems messing up our relationship with Him. Peace, or not being at odds with God, pictures that there is no friction, discomfort, or irritations in our relationship. It's a good, unhindered relationship free from problems.

These things come from God, who is by His very nature our Father (no definite article). It is quite a miracle that we can be true children of God. He wants to have a relationship with us, and be our Daddy. He wants to give us good things that we don't deserve.

These good things also come from the anointed One (Christ), whose name means Yahweh saves (Jesus), who is the Lord our. He's the boss. And He wants to use his Lordship over us to give us all good things. Obedience to Him brings about only good results.

These verses show that Paul really believes that God is a rewarder of those that diligently seek Him. 1st 2 verses = He's excited and he sees God as the giver of good things.