Servant Jesus

Peter and John were released. They returned and reported to their friends "τοὺς ἰδίους their own people" what the chief priests and elders had said (Acts 4:23). They lifted up their voice to praise God, quoting Psalm 2:1-2. They understood the whole Jesus event as a fulfillment of this prophecy. Jesus was the Messiah whom kings and rulers raised up against. Understood in this way, the Messiah carried out God's will and thus he was God's servant. No wonder in the second half of their prayer, they called Jesus "thy holy servant τὸν ἅγιον παῖδά σου" (Acts 4:27, 30).

Jesus bore several titles in the book of Acts. For example, He was
Lord Jesus ὁ κύριος Ἰησοῦς (Acts 1:21; 4:33; 7:59; 8:16; 9:17; 11:20; 15:11)
Jesus Christ Ἰησοῦ Χριστοῦ (Acts 2:38; 3:6; 4:10; 5:42; 8:12; 9:22, 34; 10:36, 48; 16:18)
(his) servant Jesus τὸν παῖδα αὐτοῦ Ἰησοῦν (Acts 3:13; 4:27, 30)
Lord Jesus Christ τὸν κύριον Ἰησοῦν Χριστόν (Acts 11:17; 15:26) etc.
Each title carries a different theology. The title "Servant" suggests that Christians at that time had not yet recognized that Jesus was God. The concept of the Blessed Trinity as we know it today had not yet been fully developed.

In the New Testament, only Luke uses this word "παῖδα, παιδός": servant Israel (Luke 1:54), servant David (Luke 1:69), the child (Luke 8:51), the boy (Luke 9:42), servant Jesus (Acts 3:13, 26; 4:25, 27, 30), the lad (Acts 20:12). The more commonly used word is "slave/servant δοῦλος" which occurs 126 times in 118 verses among which Luke-Acts takes up 29 times in 28 verses and none of them refers to Jesus.
Luke's choice of the word "παῖδα, παιδός" was ambiguous. It means a servant as well as a child (Jairus' daughter, epileptic boy) or a young man. Perhaps this is a play of word, suggesting that Jesus was the Son and Servant of God.

Here is the last part of their prayer.
And now, Lord, look upon their threats, and grant to thy servants to speak thy word with all boldness,
while thou stretchest out thy hand to heal, and signs and wonders are performed through the name of thy holy servant Jesus (Acts 4:29-30).
I am sure by now, you will be able to tell which "servant" means servant, which "servant" means son.
The Christians did not ask God to remove the threats, obstacles or persecutions. Rather, they asked God to strengthen them so that they would continue to proclaim the good news with boldness. Christians do not run away from sufferings. They embrace sufferings with boldness.
God answered them by giving all present the Holy Spirit again.
And when they had prayed, the place in which they were gathered together was shaken; and they were all filled with the Holy Spirit and spoke the word of God with boldness (Acts 4:31).
We call this the "Second Pentecost".

Dear Lord, we confess that You are the Christ, the Son of God. May we follow the example of the first believers to bear witness to Your resurrection with boldness. Amen.

Appendix:
Why did the Gentiles rage, and the peoples imagine vain things?
The kings of the earth set themselves in array, and the rulers were gathered together, against the Lord and against his Anointed
ἱνατί ἐφρύαξαν ἔθνη καὶ λαοὶ ἐμελέτησαν κενά;
παρέστησαν οἱ βασιλεῖς τῆς γῆς καὶ οἱ ἄρχοντες συνήχθησαν ἐπὶ τὸ αὐτὸ κατὰ τοῦ κυρίου καὶ κατὰ τοῦ χριστοῦ αὐτοῦ.
(Acts 4:25b-26, Psalm 2:1-2 LXX)