If I understand your question correctly, all you wish to do is to insert a hard-coded option at the top of the list, BEFORE the dynamically generated choices.
Just do this:
echo "<td><select id='bids_id' class='ui-widget-content ui-corner-all normalselect' style='width: 150px'>";
echo "<option value = '0'>Choose One</option>";
$sql = "SELECT etc etc etc
etc.
Note that you do not need to specify the selected attribute on the first, default, option tag (the "Choose One" option), because it will automatically be the selected option IF nothing else is selected. I believe that is what you would want.
What I would do, though, is to assign all values to a variable and then echo out the variable, so that everything appears all-at-once. Like this:
$opt = "<td><select id='bids_id' class='ui-widget-content ui-corner-all normalselect' style='width: 150px'>";
$opt .= "<option value = '0'>Choose One</option>";
$sql = "SELECT etc etc etc";
$querybids = $db->query($sql);
while($row = $querybids->fetch_object()) {
$opt .= "<option value='{$row->id}' ". ($row->id == $timeinfo->relatedbid ? "selected=selected" : "") .">{$row->name}</option>";
}
echo $opt;