‘Now among these men were, of the children of Judah, Daniel, Hananiah, Mishael and Azariah. And the prince of the palace servaants, (nobles, chief officers, eunuchs) gave them names, to Daniel he gave the name of Belteshazzar, and to Hananiah of Shadrach, and to Mishael of Meshach, and to Azariah of Abednego.'
The new name was given to them to bring home to them that they were now Babylonians and to give them a new status, and were now servants of the gods of Babylon. They had been ‘adopted' by the court and their future lay with the king. Giving them names connected with the gods of Babylon was intended to be a compliment. The original names meant something like, - Daniel (‘El (God) has judged'), Hananiah (‘Yahweh has been gracious'), Mishael (‘who is as El (God)'), Azariah (‘Yahweh has helped') - although we must not be over-dogmatic about the meaning of names. All were connected with the God of Israel.
The new names were connected with Babylonian thought. Daniel's with Bel. See Daniel 4:8. Some think his name was Belti - sar - usur - ‘may the lady (wife of Bel) protect the king'. Others that it was possibly only so by sound, for they see the name as signifying ‘protect his life' - balatusu-usur - but that is how names were used. It was probably intended to signify ‘Bel protect his life'). Hananiah's with Marduk (of which Shadrach was a deliberate corruption) and Azariah's with Nebo (Nego being a deliberate corruption. The name was probably intended to suggest ‘servant of Nebo'). Meshach is unidentified, it may be a deliberate corruption of Sheshach, a cypher for Babylon. Playing with names was popular amongst all cultures. Compare the sons of Jacob whose names were all given as suggestive of some idea by a play on words (Genesis 29:31 to Genesis 30:24)
Theoretically these men had now been taken from Yahweh and given to the gods of Babylon. The Babylonians were soon to be disillusioned.