How to make a Custom Field available as View Relationship?
I'm trying to have a filter that shows content where the name of
  logged in user is equal to the value of custom field.


Add your custom field as a Contextual Filter. 

Select Provide a default value then for Type select PHP Code

PHP Contextual Filter Code

  global $user; // gets current logged in user object

  if (isset($user-&gt;name)) {      
     return $user-&gt;name; // gets user name
  }