jQuery
$(document).ready(function(e){
$(".DropdownWrapper").focusin(function(e){
    $(".Dropdown").css('display', 'block');
});
$(".Item").click(function(e){
    var val = $(this).text();
    $(".Textfield").val(val);
    $(".Dropdown").css('display', 'none');
});
});
HTML
<div class="Content">
<h1>Custom Dropdown Menu For a Textfield By Farris</h1>
<div class="DropdownWrapper">
    <input type="text" class="Textfield" placeholder="Animal" />
    <div class="Dropdown">
        <div class="Item">Lion</div>
        <div class="Item">Cat</div>
        <div class="Item">Dog</div>
        <div class="Item">Monkey</div>
    </div>
</div>
</div>
CSS
body{
    background-color: #222;
    color: #FFF;
    font-family: sans-serif;
    margin: 0px;
}
.Content{
    width: 800px;
    height: auto;
    overflow: visible;
    margin: 30px auto;
}
.Textfield{
    width: 220px;
    height: 20px;
    background-color: #111;
    color: #FFF;
    font-family: sans-serif;
    padding: 15px;
    border: none;
    outline: none;
    border-radius: 3px;
}
.DropdownWrapper{
    width: 250px;
    height: 50px;
    position: relative;
    margin-bottom: 10px;
}
.Dropdown{
    position: absolute;
    top: 50px;
    left: 0px;
    width: 250px;
    height: auto;
    overflow: visible;
    z-index: 1;
    display: none;
}
.Item{
    width: 220px;
    height: 20px;
    padding: 15px;
    color: #CCC;
    background-color: #111;
}
.Item:hover{
    background-color: #000;
    cursor: pointer;
    color: #FFF;
}
This snippet will work on textfields and buttons