Hey Dan -
As you surmised, doing anything like you describe is completely unsupported and not something anything from Tableau can really help you with.
That being said, freeware bulk search/replace tools are a dime a dozen. I bet you could find something that could do this work for you in a few minutes of searching - my advice would be to back EVERYTHING up before you start playing, however
You might find this useful: http://tableaufriction.blogspot.com/2012/12/changing-database-server-name-in.html
It's a Ruby script I wrote to change the server name for data connections in a set of Workbooks.
It's not quite exactly what you were looking for - changing the name in saved data sources, but I might take a stab at that next.
Thanks Russell and Chris!