AutoComplete in jQuery with dynamically added elements

How about using another approach: initialize the autocomplete when you create the input:

$(function() {

  // settings for each autocomplete
  var autocompleteOptions = {
    minLength: 3,
    source: function(request, response) {
      $.ajax({
        type: "GET",
        url: "getNames.html",
        data: { name: request.term },
        success: function(data) {
          response(data);
        }
      });
    }
  };

  // dynamically create an input and initialize autocomplete on it
  function addInput() {
    var $input = $("<input>", {
      name: "search",
      "class": "searchInput",
      maxlength: "20"
    });
    $input
      .appendTo("form#myForm")
      .focus()
      .autocomplete(autocompleteOptions);
  };

  // initialize autocomplete on first input
  $("input.searchInput").autocomplete(autocompleteOptions);
  $("input#addButton").click(addInput);
});

<form id="myForm" name="myForm" method="post">
  <input id="addButton" type="button" value="Add an input" />
  <input name="search" class="searchInput" maxlength="20" />
</form>

jsFiddle with AJAX