What is the name that he would declare? Since this Psalm is so closely entwined with Jesus' crucifixion, the first thought that came to mind were the words that Jesus spoke to Mary after his resurrection: "...go to My brethren and say to them, 'I am ascending to My Father and your Father, and to My God and your God.'" (John 20:17) Is this the name that Psalm 22:22 points toward? Paul says that the Holy Spirit inspires Christians to address God as "Father" in prayer (Galatians 4:6) and Jesus taught believers to pray to "our Father" (Matthew 6:9). What does the rest of Scripture have to say about this?
In the Old Testament, I didn't find many occurrences of the words "God" and "Father" appearing together in reference to God. There is one occasion where David declares, "Blessed are You, LORD God of Israel, our Father, forever and ever" (1 Chronicles 29:10). In addition to this, there are two Messianic passages:
"He shall cry to Me, 'You are my Father, My God, and the rock of my salvation.'" Psalm 89:26
"... And His name will be called ... Mighty God, Everlasting Father..." Isaiah 9:6
Besides these references, there are not many other OT verses which speak like this. In fact, when Jesus called God his Father, "the Jews sought all the more to kill Him" because Jesus "said that God was His Father, making Himself equal with God" (John 5:18). Yet throughout the New Testament letters, we find over and over again the familiar greeting of Paul "Grace to you and peace from God our Father and the Lord Jesus Christ." He uses this greeting in his letters to the Christians in Rome, Corinth, Ephesus, Philippi, Colossea, and Thessaloniki and to his friend Philemon. We also see the phrase "the God and Father of our Lord Jesus Christ" used in six places by both Paul and Peter (one of the 12 apostles). Paul and James (the brother of Jesus) speak of "our God and Father" and John (one of the 12 apostles) and Jude speak of "God the Father." From these, it's clear that the NT teaches something radical about the believer's relationship with God through Christ.
Is God the Father the Person of the Trinity who is being referred to by the words "Your" in Psalm 22:22? Is the relationship of Father-child the primary way that a Christian is to approach God in Christ - the "name" by which he desires to be known? These are all deep things that require more meditation and study. It seems that it would do me well to continue digging into the word in order to find the God who is, who was, and who is to come.
"...Jesus Christ, the faithful witness, the firstborn from the dead, and the ruler over the kings of the earth. To Him who loved us and washed us from our sins in His own blood, and has made us kings and priests to His God and Father, to Him be glory and dominion forever and ever. Amen." Revelation 1:5-6
God our Father - Romans 1:7, 1 Corinthians 1:3, 1 Corinthians 1:2, Ephesians 1:2, Philippians 1:2, Colossians 1:2, 1 Thessalonians 1:1, 2 Thessalonians 1:1, 2 Thessalonians 1:2, 1 Timothy 1:2, Philemon 1:3
God the Father - 1 Corinthians 8:6, 1 Corinthians 15:24, Galatians 1:1, Galatians 1:3, Ephesians 5:20, Ephesians 6:23, Phil 2:11, Colossians 3:17, 1 Thessalonians 1:1, 2 Timothy 1:2, Titus 1:4, 1 Peter 1:2, 2 Peter 1:17, 2 John 1:3, Jude 1:1
our God and Father - Galatians 1:4, Philippians 4:20, 1 Thessalonians 1:3, 1 Thessalonians 3:11, 1 Thessalonians 3:13, 2 Thessalonians 2:16, James 3:9
the God and Father of our Lord Jesus Christ - Romans 15:6, 2 Corinthians 1:3, 2 Corinthians 11:31, Ephesians 1:3, Colossians 1:3, 1 Peter 1:3