Adoption is an analogy describing the new relationship we have in Christ. We cannot be in God’s family at birth! After we are “born again,” we are in God’s family. A person must want to be in God’s family to make the transition spiritually out of Satan’s domain (Jn. 3:1–21). At times, the term “Sons of God” is used independently or in conjunction with “adoption.” We borrow the definitions as follows:
ADOPTION (υἱοθεσία, huiothesia). A Greek phrase that describes the act of raising a child who is not biologically related. Used by Paul to describe God adopting humans — implied elsewhere in the Bible to represent membership in the family of God.
Adoption. Theologically, the act of God by which believers become members of “God’s family” with all the privileges and obligations of a family membership. “Sons of God,” a common KJV expression, includes individuals of both sexes numbered among God’s children (Is 43:6; 2 Cor 6:18). 
Some key verses:
The text explains that Jesus (vs. 10) arrived in this world as the Messiah, but the Jews did not accept Jesus in this role. Jesus fulfilled all the required prophesies for First Coming but rejected nonetheless; First by the religious establishment, then by Israel. The Jewish society was already in disobedience and under the rule of Satan. Thus, the sinful nature of the Jewish nation blinded the people and kept them in an ignorant state. This rejection was predicted in Isa. 53:1. The rejection by Israel makes possible the Church Age and a postponement of the Messianic Kingdom. Israel will receive the offer again during the Tribulation. The postponement is from our perspective, every thing that happens is in God’s plan and not a surprise to Him (Is. 52:13–53:12, Is. 49:1–13). Paul laments that he cannot take on the punishment for Israel’s sin (Rom 9:3) [4,5]
Some did accept Jesus as the Messiah. Those that believed the truth about Jesus as the Messiah now have the rights as “Sons of God.” Jesus Christ is the one son of God before the creation of the universe and everything in it, visible and invisible. Now, as legitimate children, we have those legal rights. Salvation provides adoption as one of the immediate benefits we receive at Salvation. Believing in the “name” of Jesus means:
“To be a child of God means to accept the incredible reality of the coming of the Logos [Jesus Christ] personally into the human situation. To know and believe personally that the power of the universe can touch your life means that one can be adopted into the family of the Logos and become a child of God. “
Verse 13 uses contrast about the nature of becoming a Child of God. A person’s decision to accept Jesus as their Saviour does not involve “will power” or location of birth. For example, those born in Israel are not automatically Christians. This rebirth is Spiritual, hence Old Man/ New Man usage in Scripture.
Verses 12–13 in this passage describes the mechanism by which we become Children of God. The term adoption is not here but implied with the “Children of God.” The way to become one of those children by the work of the Holy Spirit. Believers receive this Spirit the moment of Salvation described above. As adopted children, we share in the blessing and sufferings that come with the Salvation event. The purpose is to glorify the Lord.
Some of the benefits of adoption come at the Second Coming of Christ. Christ will summon believers, dead and alive, before His return to complete the adoption. Until that time, we have the indwelling of the Holy Spirit as a guarantee (Eph. 1:13–14).
Hopefully, you did not miss the fact that creation waits for this event to complete. Verse 18 indicates that whatever we endure during this life, pails to the glory to be revealed latter.
1. Morris, M. J. (2016). Adoption. In J. D. Barry, D. Bomar, D. R. Brown, R. Klippenstein, D. Mangum, C. Sinclair Wolcott, … W. Widder (Eds.), The Lexham Bible Dictionary. Bellingham, WA: Lexham Press.
2. Shepherd, N. (1988). Adoption. In Baker encyclopedia of the Bible (Vol. 1, p. 31). Grand Rapids, MI: Baker Book House.2.
3. Blum, E. A. (1985). John. In J. F. Walvoord & R. B. Zuck (Eds.), The Bible Knowledge Commentary: An Exposition of the Scriptures (Vol. 2, p. 272). Wheaton, IL: Victor Books.
4. Borchert, G. L. (1996). John 1–11 (Vol. 25A, p. 114). Nashville: Broadman & Holman Publishers.
5. Fruchtenbaum, A. G. (1983). The Messianic Bible Study Collection (Vol. 182, pp. 16–17). Tustin, CA: Ariel Ministries.8
6. Borchert, G. L. (1996). John 1–11 (Vol. 25A, pp. 116–117). Nashville: Broadman & Holman Publishers.
7. Borchert, G. L. (1996). John 1–11 (Vol. 25A, p. 118). Nashville: Broadman & Holman Publishers.