How to override default System.Resources.ResourceManager in Resources.Designer.cs?

I believe you're asking two separate questions here. You can certainly override ResourceManager.GetString. You can't, however, use that override in the auto-generated Resource.Designer.cs code. To use it, you'll have to write your own Resource designer class.

public class MyResourceManager : System.Resources.ResourceManager
{
  // override
  public override GetString(string name)
  {
    // custom code
  }
}

public class MyResourceDesigner
{
  // use your custom class with override
  private static MyResourceManager resourceManager;
  private static CultureInfo resourceCulture;

  public static MyResourceManager ResourceManager
  {
    get
    {
      if (object.ReferenceEquals(resourceManager, null))
      {
        // Resource is just the name of the .resx file
        // be sure to include relevant namespaces
        var temp = new MyResourceManager(
            "MyProject.Resource", 
            typeof(MyResourceDesigner).Assembly);
        resourceManager = temp;
      }

      return resourceManager;
    }
  }

  public static CultureInfo Culture
  {
    get
    {
      return resourceCulture;
    }

    set
    {
      resourceCulture = value;
    }
  }

  // start adding strongly-typed objects
  public static string Foo
  {
    get
    {
      // use your override
      return ResourceManager.GetString("Foo", resourceCulture);
    }
  }
}

I created a CustomResourceManager that overrides the GetString() calls like so:

public class CustomResourceManager : ResourceManager
{
    public CustomResourceManager(Type resourceSource)
        :base(resourceSource)
    {
    }

    public CustomResourceManager(string baseName, Assembly assembly)
        : base(baseName, assembly)
    {
    }

    public CustomResourceManager(string baseName, Assembly assembly, Type usingResourceSet)
        : base(baseName, assembly, usingResourceSet)
    {
    }

    public override string GetString(string name)
    {
        // your business logic
    }

    public override string GetString(string name, CultureInfo culture)
    {
        // your business logic
    }
}

Then I added a pre-build step to my project to replace the creation of a System.Resources.ResourceManager with my CustomResourceManager in the generated Resource.Designer.cs file:

powershell -command "(gc ..\Resources\Resource.Designer.cs).Replace(\"new global::System.Resources.ResourceManager\", \"new CustomResourceManager\") | set-content ..\Resources\Resource.Designer.cs -Encoding UTF8"

And away we go!