jQuery hasAttr checking to see if there is an attribute on an element

var attr = $(this).attr('name');

// For some browsers, `attr` is undefined; for others,
// `attr` is false.  Check for both.
if (typeof attr !== 'undefined' && attr !== false) {
    // ...
}

If you will be checking the existence of attributes frequently, I would suggest creating a hasAttr function, to use as you hypothesized in your question:

$.fn.hasAttr = function(name) {  
   return this.attr(name) !== undefined;
};

$(document).ready(function() {
    if($('.edit').hasAttr('id')) {
        alert('true');
    } else {
        alert('false');
    }
});

<div class="edit" id="div_1">Test field</div>

How about just $(this).is("[name]")?

The [attr] syntax is the CSS selector for an element with an attribute attr, and .is() checks if the element it is called on matches the given CSS selector.

You're so close it's crazy.

if($(this).attr("name"))

There's no hasAttr but hitting an attribute by name will just return undefined if it doesn't exist.

This is why the below works. If you remove the name attribute from #heading the second alert will fire.

Update: As per the comments, the below will ONLY work if the attribute is present AND is set to something not if the attribute is there but empty

<script type="text/javascript">
$(document).ready(function()
{
    if ($("#heading").attr("name"))
      alert('Look, this is showing because it\'s not undefined');
    else
      alert('This would be called if it were undefined or is there but empty');
});
</script>
<h1 id="heading" name="bob">Welcome!</h1>