I assume that you have set image's build action to Resource, which is default value. And your string value must be like that:
string addImage = "/Application;component/Images/image_name.png";
I all the above is OK, then the problem must be in your Button
. You have set it's DataContext to ListBox1
DataContext. Why? There is no need.
Change
<Button Height="80" Width="80" DataContext="{Binding DataContext, ElementName=listBox1}" Command="{Binding addPerson}">
<Button.Background>
<ImageBrush ImageSource="{Binding addImage}" Stretch="Fill" />
</Button.Background>
</Button>
to
<Button Height="80" Width="80" Command="{Binding addPerson}">
<Button.Background>
<ImageBrush ImageSource="{Binding addImage}" Stretch="Fill" />
</Button.Background>
</Button>