How do you return a menu with the menu name?

If you want to return the output of wp_nav_menu(), instead of echo-ing it, you should replace

'echo'            => 1,

with

'echo'            => 0,

Then you should get something out of var_dump( $menu );.

You can read more about the input parameters in the Codex.

Update:

You should remove this line:

'items_wrap'      => '',

to use the default which is:

'items_wrap'      => '<ul id="%1$s" class="%2$s">%3$s</ul>',

You can also target a specific menu with the 'menu' parameter, if you haven’t selected a location for it, in the menu settings page. Currently you have it set to '';