The dropdown box that you are talking about is for custom WordPress template that WordPress supports by default.
In order to override woocomerce template e.g. single-product.php
what you have to do is to create a folder with name woocommerce
inside your theme folder e.g.
/wp-content/themes/{your_theme_name}/woocommerce/
and copy the file from /wp-content/plugins/woocommerce/templates/single-product.php
into your theme’s woocommerce directory e.g.
/wp-content/themes/{your_theme_name}/woocommerce/single-product.php
Now its properly override and you can make changes in single-product.php
within your theme.
This way your changes will remain intact even if you update woocommerce plugin.
To know in detail about woocomerce template overriding see this link
Edit:
Declaring WooCommerce support in themes
Declaring WooCommerce support is straightforward and involves adding one function in your theme’s functions.php file.
function mytheme_add_woocommerce_support() {
add_theme_support( 'woocommerce' );
}
add_action( 'after_setup_theme', 'mytheme_add_woocommerce_support' );