Pass JavaScript value to PHP using JQUERY Ajax (Values are from Facebook Javascript SDK)

I'm not 100% sure I understand your question, but what it seems like you're saying is that once you get the the point where you want to submit something back to the server (PHP code), you want to do it asynchronously via jQuery's $.ajax() method? If so, you can still use a submit button without refreshing the page. Just wire up an event listener to the button to make an AJAX request when it is clicked

HTML:

<form id="theForm">
    <input type="text" id="textBox" name="textBox" value="textBox" />
    <input type="text" id="userName" name="userName" value="userName" />
    <input type="button" id="submitBtn" value="Submit" />
</form>

JS:

$('#submitBtn').on('click', function() {
    var formData = $('#theForm').serialize();
    $.ajax({
        url: 'something.php',
        data: formData,
        success: function (data) {},
        error: function() {}
    })
});

Even better, if you want your data serialized into an array of objects rather than a query string, you can use .serializeArray() instead. But PHP should be able to read all the values from the query string as long as you have names associated with your form elements. Then whatever you send back from PHP will be contained in the data variable of your success function. And this will all execute without a page refresh.

If you don't want to use a button at all, I supposed you could use setTimeout() so it would just send an AJAX request every X milliseconds or somehow check if changes are made on your page, but that could get pretty complicated. I would say keep it simple at this point.

EDIT: After getting more information from your comment, the answer is even simpler than above. Still, all you're doing is making an AJAX request inside your FB.api() function:

FB.api(uid, function(info) {    
    $('#welcome').html("Hello dear " + info.name );    
    $.ajax({
        url: 'something.php',
        data: info,
        success: ... ,
        error: ...
    });
});

From there, you just grab the variable from the form variable in PHP. My PHP is a little rusty, but I believe it would be something like $_POST["info"]. Trying to redisplay it without refreshing the page is completely pointless because you already have the variable in the page. The only point of sending the username off to the server is to save it in the database. You've already displayed it on the page through the API.