You can use absolute positioning and transform on the img while setting the parent container (inner-div) to relative position:
   position:absolute; /* Absolute relative to inner-div */
   left:50%;   /* Top, left and transform for horizontal and vertical alignment without the use of margin or changing the display */
   top:50%;
   transform: translate(-50%,-50%);
Fiddle : https://jsfiddle.net/7t2fghtf/
$("#mybutton").click(function() {
  //  $.ajax("").done(function(){
  //          $("#mycontent").html( $("#anothermymodal").html()); 
  //          })
  setTimeout(function() {
    $("#mycontent").html($("#anothermymodal").html());
  }, 10000);
});
.inner-div {
  margin: 0 auto;
  width: 100px;
  height: 100px;
  border-radius: 3px;
  display: table;
  vertical-align: middle;
  text-align: center;
  background-color: red;
  position: relative;
}
img {
  position: absolute;
  left: 50%;
  top: 50%;
  transform: translate(-50%, -50%);
  display: inline-block;
  margin-left: auto;
  margin-right: auto;
}
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css">
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js"></script>
<div class="container">
  <h2>Small Modal</h2>
  <!-- Trigger the modal with a button -->
  <button type="button" id="mybutton" class="btn btn-info btn-lg" data-toggle="modal" data-target="#myModal">Open Small Modal</button>
  <div style="display:none" id="anothermymodal">
    <div class="modal-body">
      <p>This is a small modal.</p>
    </div>
    <div class="modal-footer">
      <button type="button" class="btn btn-default" data-dismiss="modal">Close</button>
    </div>
  </div>
</div>
<!-- Modal -->
<div class="modal fade" id="myModal" role="dialog">
  <div class="modal-dialog">
    <div class="modal-content" id="mycontent">
      <div class="inner-div">
        <img src="http://www.ajaxload.info/cache/FF/FF/FF/00/00/00/5-1.gif" />
      </div>
    </div>
  </div>
</div>