🔼The name Sheshach in the Bible
The name Sheshach (or Sheshak) occurs twice in the Bible, both times in the Book of Jeremiah. In Jeremiah 25 the prophet tells about a cup of the "wine of wrath" he takes from the hand of YHWH and makes all the nations drink to whom God sent him (25:15). He submits a long list of pretty much every known nation but then, as an afterthought, he adds the king of Sheshach in verse 26.
In Jeremiah 51:41, the prophet exclaims a vision of a future in which Sheshach has been captured and Babylon has become an object of horror. It's obvious that Sheshach has something to do with Babylon, but what precisely isn't clear.
Perhaps Sheshach was a little known region or town within the empire, back in the day known for something outstanding, but more likely Sheshach is a pseudonym for Babylon. Why Babylon would need a pseudonym (or several, because Leb-kamai appears to be another one) is also not very clear, but Sheshach (ששך) can easily be constructed from Babel (בבל) via the Athbash cypher (replacing the letter א with the ת, the letter ב with the ש, and so on).
🔼Etymology of the name Sheshach
Because Sheshach is probably an artificially constructed word it has no proper etymology. Still, one can't help but wonder what the first Jewish readers of this word thought it meant until they figured out it was Babel scrambled. Much of the Hebrew texts and Christian Greek texts we have come from a period of invasions and great civil unrest, and were produced from the side of the resistance. It's very well possible that they contain vastly complex references and encryptions that we simply don't know about (yet).
The name Sheshak looks like it's any of the words שש suffixed with the ך, which may mark the second person possessive form: your שש:
The name Sheshach probably simply means Babel, but it remains a mystery why Jeremiah alternated his use of the proper name Babel with Sheshach. The name Sheshach may also mean Your Lilly / Six / Alabaster, etcetera.