You can find a YouTube playlist here with many of the songs suggested below.
The psalm can suggest opening with songs of praise to the Lord who is our king (also a potential link to the Gospel) and we are called to praise God’s great and awesome name. Holy, holy, holy, Lord God almighty (CH 111 / MP 237), Immortal, invisible, God only wise (CH 132 / MP 327) and How great thou art (CH 154 / MP 506) would some classic hymn options while more recent songs to consider include Come people of the risen King (MP 1267 / CCLI / Getty), Majesty (MP 454 / CCLI), High praise (CCLI) or Come and worship Christ the King (Resound).
The Lord is king enthroned in might (PFAS 99A / Hymnary) sung to Ellacombe and God reigns, enthroned on high (Hope) sung to Leoni are both options for singing the psalm metrically to well known tunes. He is exalted (CH 437 / MP 217) would be a good alternative as a chorus which picks up the key theme of the psalm while the chorus of Exalt the Lord our God (Satellite) could potentially be used for responsorial singing.
Baited, the question rose (Hope) sung to Diademata and Is it lawful to pay taxes (Carolyn) sung to Beach Spring are two of the few songs which specifically link to the Gospel story. Lord, you are more precious than silver (MP 447 / CCLI) contrasts the value we place on precious things and the value we place on God while O God, we bear the imprint of your face (CH 254) explores us being made in God’s image if you are reflecting on us being one of the things that are God’s.
Rejoice in God’s saints (CH 742) or As stars adorn the night-veiled sky (CH 748) could be a suitable response to the Epistle as we celebrate all those like the church of the Thessalonians who have received the Gospel and lived it out. You could also think about the unity shown through this with I am the Church! You are the Church (CH 204), We are one in the Spirit (Hymnary) or Dear friends, we’re one (STB 311 / Hymnary) from Laos, the latter particularly emphasising the global reach of the Christian message.
As God promises to proclaim their name to Moses in Exodus you could use songs which explore what we call God such as Bring many names (CH 134) or Source and Sovereign, Rock and Cloud (CH 133) does the incredible task of giving twelve names for each member of the Trinity set to the majestic tune Aberystwyth. You could also consider Blessed be your name (MP 1036 / CCLI), Your name (MP 1385 / CCLI) and Exalt the Lord our God (Satellite) which focus more on singing praise for God’s name.
You could finish with songs which proclaim the majesty of God or which celebrate the name of God, including How shall I sing that majesty (CH 128), All hail the power of jesus’ name (CH 457 / MP 13), At the name of Jesus (CH 458 / MP 41), Meekness and majesty (CH 356 / MP 465), Lord, I lift your name on high (CH 558 / MP 881), Blessed be your name (MP 1036 / CCLI) and O praise the name (Anástasis) (CCLI).