How to change the MenuItem Header according to the value selected in its submenu?

Question

Try this code:

XAML

<Grid>       
    <Grid.Resources>
        <Style x:Key="MyMenuItem" 
               BasedOn="{StaticResource {x:Type MenuItem}}"
               TargetType="{x:Type MenuItem}">

            <EventSetter Event="Click" Handler="headerLogoutTime_Click" />
            <EventSetter Event="MouseEnter" Handler="headerMin_MouseEnter" />
            <Setter Property="Background" Value="#FF004165" />
        </Style>
    </Grid.Resources>

    <Menu Height="23" Name="menu1" Width="171" Margin="10,10,0,0" HorizontalAlignment="Center" VerticalAlignment="Top" Background="#FF004165">
        <MenuItem Header="Settings">
            <MenuItem x:Name="headerLogoutTime"
                      Header="LogOut Time: 00 Min" 
                      Background="#FF004165">

                <MenuItem x:Name="header10Min" 
                          Header="10 Min"
                          Style="{StaticResource MyMenuItem}" />

                <MenuItem x:Name="header20Min"
                          Header="20 Min"
                          Style="{StaticResource MyMenuItem}" />

                <MenuItem x:Name="header30Min"
                          Header="30 Min" 
                          Style="{StaticResource MyMenuItem}" />

                <MenuItem x:Name="header40Min"
                          Header="40 Min" 
                          Style="{StaticResource MyMenuItem}" />

                <MenuItem x:Name="header50Min" 
                          Header="50 Min"
                          Style="{StaticResource MyMenuItem}" />
            </MenuItem>

            <MenuItem Header="LogOut" Background="#FF004165" />
        </MenuItem>
    </Menu>
</Grid>

Code-behind

public partial class MainWindow : Window
{
    public MainWindow()
    {
        InitializeComponent();
    }

    private void headerLogoutTime_Click(object sender, RoutedEventArgs e)
    {
        SetMenuItemHeader(sender as MenuItem);
    }

    private void headerMin_MouseEnter(object sender, MouseEventArgs e)
    {
        SetMenuItemHeader(sender as MenuItem);
    }

    private void SetMenuItemHeader(MenuItem menuItem) 
    {
        var sb = new StringBuilder();

        if (menuItem != null)
        {
            headerLogoutTime.Header = "";
            headerLogoutTime.Header = sb.Append("LogOut Time: ").Append(headerLogoutTime.Header).Append(menuItem.Header);
        }
    }
}

Clarification

In MyMenuItem Style we define a Click and MouseEnter handlers for all MenuItems, what contains in headerLogoutTime, here it also we set the Background color.

In headerLogoutTime_Click and in headerMin_MouseEnter handlers, with help of StringBuilder setting new value for MenuItem Header. Class StringBuilder chosen to enhance performance for working with strings.

Edit: add line as separator

To display the line, as in your picture, use Separator control with Style:

<Style x:Key="SeparatorStyleKey" 
        TargetType="{x:Type Separator}">

    <Setter Property="OverridesDefaultStyle" Value="True" />
    <Setter Property="SnapsToDevicePixels" Value="True" />
    <Setter Property="Background" Value="White" />
    <Setter Property="Height" Value="1" />

    <Setter Property="Template">
        <Setter.Value>
            <ControlTemplate TargetType="{x:Type Separator}">
                <Rectangle SnapsToDevicePixels="{TemplateBinding SnapsToDevicePixels}"
                           Height="{TemplateBinding Height}"
                           Width="{TemplateBinding Width}"
                           Fill="{TemplateBinding Background}" />
            </ControlTemplate>
        </Setter.Value>
    </Setter>
</Style>

And use like this:

<MenuItem Header="Settings">
    <MenuItem x:Name="headerLogoutTime"
                Header="LogOut Time: 00 Min" 
                Background="#FF004165">

        <MenuItem x:Name="header10Min" 
                    Header="10 Min"
                    Style="{StaticResource MyMenuItem}" />

        <Separator Style="{StaticResource SeparatorStyleKey}" />

        <MenuItem x:Name="header20Min"
                    Header="20 Min"
                    Style="{StaticResource MyMenuItem}" />

        <Separator Style="{StaticResource SeparatorStyleKey}" />

        <MenuItem x:Name="header30Min"
                    Header="30 Min" 
                    Style="{StaticResource MyMenuItem}" />

        <Separator Style="{StaticResource SeparatorStyleKey}" />

        <MenuItem x:Name="header40Min"
                    Header="40 Min" 
                    Style="{StaticResource MyMenuItem}" />

        <Separator Style="{StaticResource SeparatorStyleKey}" />

        <MenuItem x:Name="header50Min" 
                    Header="50 Min"
                    Style="{StaticResource MyMenuItem}" />
    </MenuItem>

    <MenuItem Header="LogOut" Background="#FF004165" />
</MenuItem>