If you want to avoid users updating their database connections, then I think the only way to do this would be to:
1) Create a new server with the same name (but perhaps not on the same network/domain initially).
2) Install SQL Server and use the same instance name.
3) Back up and restore the database from the old server to the new server.
4) Optionally upgrade the geodatabase to the latest version (your backup will still be the same geodatabase. Version as the old system - i.e. 10.1)
5) Take the old server off the network/domain and then add the new server to the network/domain
As you say, the server name is embedded in the connection, so if the new server does not have the same name, then the old connections won't work.
John