By default, the report connections list is populated based on the connections stored in the application configuration file. In any case,, the application configuration cannot be modified at runtime, that's why the report connections list is not changed when the Reports End-User Designer is used.
To initialize, save and restore the connection strings in the Reports End-User Designer, use the approach demonstrated in this code example.
The main idea of this approach is to create a custom connection storage service (the ConnectionStorageService class descendant). Also, the SqlDataConnection class descendant (CustomSqlDataConnection) is used in this example to customize the connection names in the connections list.