I had no luck searching for the origin of "getting thrown in the clink" on your website. I did find the following on the taboo website whose origin (URL) I will not mention here:
The name of a prison which was on Clink Street in the Southwark area of London.
Claudie wrote: I always thought that the doors of the cells went "clink" when they shut behind the prisoner. Maybe a more accurate name would be the "clang".
I had heard the second (clink/clang) meaning and origin but not the first. Do you know what the origin is?