I'm not a PHP developer, so I only created a dirty fix for this:

core/layout_api.php: after if ( $t_show_project_selector) }:

?>

<span class="input-icon" style="padding-top: 4px;">

<input type="text" name="searchFor" ID="searchFor" autocomplete="off" class="nav-search-input" placeholder="Search Project" onKeyUp="searchSelect(this.id,'project_id')"

style="border-radius: 4px !important; width: 150px;">

<i class="ace-icon fa fa-search">
</span>
<?php

html_javascript_link('custom.js');

the < ul > was given an extra id tag:

echo '<ul name="project_id" class="dropdown-menu dropdown-menu-right dropdown-yellow dropdown-caret dropdown-close scrollable-menu">' . "\n";

the custom.js is as follows:

function searchSelect(searchForID, searchInID) {

var input = document.getElementById(searchForID).value.toLowerCase();
var output = document.getElementsByName(searchInID)[0];

output = output.getElementsByTagName("a");

for (var i = 0; i < output.length; i++) {
if (output[i].text.toLowerCase().indexOf(input) < 0) {
    output[i].style.display = "none";
    output[i].setAttribute("style", "display:none");

} else {
    output[i].style.display = "";
    output[i].setAttribute("style", "display:");
}
}    

var optionsPossible = 0;
var LastPossible = 0;
for (var i = 0; i < output.length; i++) {
    if (output[i].style.display == "") {
        optionsPossible += 1;
        LastPossible = output[i];
    }
}
if (optionsPossible == 1) {
    LastPossible.click();
}
}