Dropdown-menu Bootstrap with PHP

If you do this (just an example), you can send everything in a request only for page.php:

    <div class="btn-group">
    <button id="drop" class="btn btn-default dropdown-toggle" type="button" data-toggle="dropdown" aria-expanded="false">
        BOTÂO <span class="caret"></span>
    </button>                   
    <ul class="dropdown-menu" role="menu" id="status">
        <li data-something="1"><a href="#">valor 1</a></li>
        <li data-something="2"><a href="#">valor 2</a></li>
    </ul>
    <input type='hidden' id='selected_data'>
</div>
<br><br>
<div id='output'></div> 


<div class="modal-body">
<textarea id="area_1" name="area_1" class="form-control" rows="7"></textarea>
<br><br>    
<!-- BUTTON TO SEND THE VALUE AND THE TEXTAREA TO PHP PAGE-->    
<button type="button" id="enviar" class="btn btn-default">SEND TO PHP</button> 

$(document).on('click','#status li',function(){
        $('#selected_data').val($(this).data('something'));
    });

$(document).on('click','#enviar',function(){
        $.ajax({
            url: 'page.php',
            type: 'POST',
            data: {
                    valor: $('#selected_data').val(),
                    plaintext: $('#area_1').val(),                  
                  },

            success: function(resposta){
                $("#area_1").html(resposta);
            }
        }).done(function(data){
           alert("Successfull", data);          
       });
    });

Luis, the way to get the value of the attribute, as André Ribeiro has passed, is correct. The error seems to be in the way you are constructing the "option data" of JQuery.ajax. Remove the quotation marks from the "value" index, like this:

$(document).on('click','#status li',function(){
$.ajax({
    url: 'exemple.php',
    type: 'POST',
    data: {valor: $(this).attr('data')}
    });
}); 

To get the value of the data attribute from the li of your menu you can use the .attr( attributeName ) function.

In your code it would look like this:

$(document).on('click','#status li',function(){
  $.ajax({
        url: 'exemple.php',
        type: 'POST',
        data: {
               'valor': $(this).attr('data')
              },
        });
}); 

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script><divclass="btn-group">
 <button class="btn btn-danger btn-lg dropdown-toggle" type="button" data-toggle="dropdown" aria-expanded="false">
                    BOTÂO <span class="caret"></span>
                    </button>                   

                    <ul class="dropdown-menu" role="menu" id="status">
                        <li data="1"><a href="#">valor 1</a></li>
                        <li data="2"><a href="#">valor 2</a></li>
                    </ul>
</div>