How Can I Open a WPF Popup with a Delay?

You can create a style to be applied to the Popup in the following way:

<Style x:Key="TooltipPopupStyle" TargetType="Popup">
    <Style.Triggers>
        <DataTrigger Binding="{Binding PlacementTarget.IsMouseOver, RelativeSource={RelativeSource Self}}" Value="True">
            <DataTrigger.EnterActions>
                <BeginStoryboard x:Name="OpenPopupStoryBoard" >
                    <Storyboard>
                        <BooleanAnimationUsingKeyFrames Storyboard.TargetProperty="IsOpen" FillBehavior="HoldEnd">
                            <DiscreteBooleanKeyFrame KeyTime="0:0:0.25" Value="True"/>
                        </BooleanAnimationUsingKeyFrames>
                    </Storyboard>
                </BeginStoryboard>
            </DataTrigger.EnterActions>
            <DataTrigger.ExitActions>
                <PauseStoryboard BeginStoryboardName="OpenPopupStoryBoard"/>
                <BeginStoryboard x:Name="ClosePopupStoryBoard">
                    <Storyboard>
                        <BooleanAnimationUsingKeyFrames Storyboard.TargetProperty="IsOpen" FillBehavior="HoldEnd">
                            <DiscreteBooleanKeyFrame KeyTime="0:0:1" Value="False"/>
                        </BooleanAnimationUsingKeyFrames>
                    </Storyboard>
                </BeginStoryboard>
            </DataTrigger.ExitActions>
        </DataTrigger>

        <Trigger Property="IsMouseOver" Value="True">
            <Trigger.EnterActions>
                <PauseStoryboard BeginStoryboardName="ClosePopupStoryBoard" />
            </Trigger.EnterActions>
            <Trigger.ExitActions>
                <PauseStoryboard BeginStoryboardName="OpenPopupStoryBoard"/>
                <ResumeStoryboard BeginStoryboardName="ClosePopupStoryBoard" />
            </Trigger.ExitAct