Display a default DataTemplate in a ContentControl when its content is null or empty?

Simple, you have to bind the content property in the style. Styles won’t overwrite a value on a control if there’s a binding present, even if the value evaluates to Null. Try this.

<ContentControl>
    <ContentControl.Style>
        <Style TargetType="ContentControl">
            <Setter Property="Content" Value="{Binding HurfView.EditedPart}" />
            <Style.Triggers>
                <DataTrigger Binding="{Binding RelativeSource={x:Static RelativeSource.Self}, Path=Content}" Value="{x:Null}">
                    <Setter Property="ContentControl.Template">
                        <Setter.Value>
                            <ControlTemplate>
                                <Grid HorizontalAlignment="Stretch" VerticalAlignment="Stretch">
                                    <TextBlock>EMPTY!</TextBlock>
                                </Grid>
                            </ControlTemplate>
                        </Setter.Value>
                    </Setter>
                </DataTrigger>
            </Style.Triggers>
        </Style>
    </ContentControl.Style>
</ContentControl>