Here
http://en.wikipedia.org/wiki/Sin_(mythology)
is an article on the Babylonian Moon God Sin.
The ancient Hebrews were none too keen on Babylon or its faiths, for good reason.
Does the word “sin”–an immoral act that offends god—derive from this name?
No.
From the OED:
So the word is PIE, not Babylonian. It doesn’t derive from Hebrew, either, or from any language that has any connection to Babylon.
It’s a coincidence, much like the fact that there is a Catholic Church official in the Phillippines named “Cardinal Sin.”
As RealityChuck said, there’s no etymological connection between “sin” and “Sin”.
There is, however, probably a connection between “Sin” and the region known as Sinai. The Babylonians and Assyrians had a few occasions to experience the wonders of this lovely spot.