How should I edit an Entity Framework connection string?

No, you can't edit the connection string in the designer. The connection string is not part of the EDMX file it is just referenced value from the configuration file and probably because of that it is just readonly in the properties window.

Modifying configuration file is common task because you sometimes wants to make change without rebuilding the application. That is the reason why configuration files exist.

You normally define your connection strings in Web.config. After generating the edmx the connection string will get stored in the App.Config. If you want to change the connection string go to the app.config and remove all the connection strings. Now go to the edmx, right click on the designer surface, select Update model from database, choose the connection string from the dropdown, Click next, Add or Refresh (select what you want) and finish.

In the output window it will show something like this,

Generated model file: UpostDataModel.edmx. Loading metadata from the database took 00:00:00.4258157. Generating the model took 00:00:01.5623765. Added the connection string to the App.Config file.

If you remove the connection string from the app.config file, re-running the entity Data Model wizard will guide you to build a new connection.