Going to prison is called "doing a bid". What's the history behind that? Is it based on "doing bird" (based on being locked up like a bird)?
It appears that 'doing a bit' goes back to the early 20th century as a term for serving a prison sentence.
'Bit' meaning a short span of time goes back to the 17th century, and presumably that's how it became associated with prison in the first place. "I've got to do a bit of time in prison."
I can't find any real resources as to where 'bid' came from directly, so I would assume that it is simply a result of the word shifting slightly once having gained distinct enough usage separate from its origins.