<Grid
  xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
  xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
  xmlns:local="clr-namespace:Testing">
  <StackPanel>
    <local:InputActionGlyph Action="test"/> <!-- This works -->
    <local:InputActionGlyph Action="{Binding MyActionName}"/>	<!-- This doesn't work -->
    <TextBlock Text="{Binding MyActionName}"/> 	<!-- This works -->
  </StackPanel>
</Grid>