You could add a Style
for Run
to subscribe e.g. MouseLeftButtonDown
<RichTextBox.Resources>
<Style TargetType="Run">
<EventSetter Event="MouseLeftButtonDown" Handler="Run_Click" />
</Style>
</RichTextBox.Resources>
and handle the event like this
void Run_Click(object sender, MouseButtonEventArgs e)
{
Run run = sender as Run;
run.Background = Brushes.Red;
}