SSIS 2012 Project Connection Manager does not expose ServerName property for configuration in SSMS 2012
I found the answer by opening the project with Visual Studio 2012.

Open the SSIS project in Visual Studio, and select the .conmgr file in the Solution Explorer pane. Use the context menu and select the "View Code" menu option. This will show the XML within the Connection Manager file.

In the XML, there was this tag:

&lt;DTS:PropertyExpression
    DTS:Name="ServerName"&gt;@[$Project::my_server_name]&lt;/DTS:PropertyExpression&gt;


The ServerName property was set via an expression!

This fact is not easily discernible via the Solution Explorer tree itself. When the conmgr file is selected, the Properties pane only displays the File Name and Full Path properties.

When you open a package that uses the Connection Manager, and select that Connection Manager within the package editor pane, the properties pane shows many other properties, including Expressions. The ServerName property will be visible in the Expressions list only if it has been set.

There are two ways to resolve this issue:


Within the Visual Studio project, open a package. Select the Connection Manager from the package editor pane, and delete the Property Expression from the Properties pane.
Remove the PropertyExpression tag from the .conmgr file with a text editor.


Either way, once the ServerName property is no longer set by an expression within the connection manager definition file, it is exposed as a configurable parameter in the Configure dialog box in SSMS 2012.