How to make a section of an image a clickable link

If you don't want to make the button a separate image, you can use the <area> tag. This is done by using html similar to this:

<img src="imgsrc" width="imgwidth" height="imgheight" alt="alttext" usemap="#mapname">

<map name="mapname">
    <area shape="rect" coords="see note 1" href="link" alt="alttext">
</map>

Note 1: The coords=" " attribute must be formatted in this way: coords="x1,y1,x2,y2" where:

x1=top left X coordinate
y1=top left Y coordinate
x2=bottom right X coordinate
y2=bottom right Y coordinate

Note 2: The usemap="#mapname" attribute must include the #.

EDIT:

I looked at your code and added in the <map> and <area> tags where they should be. I also commented out some parts that were either overlapping the image or seemed there for no use.

<div class="flexslider">
    <ul class="slides" runat="server" id="Ul">                             
        <li class="flex-active-slide" style="background: url(&quot;images/slider-bg-1.jpg&quot;) no-repeat scroll 50% 0px transparent; width: 100%; float: left; margin-right: -100%; position: relative; display: list-item;">
            <div class="container">
                <div class="sixteen columns contain"></div>   
                <img runat="server" id="imgSlide1" style="top: 1px; right: -19px; opacity: 1;" class="item" src="./test.png" data-topimage="7%" height="358" width="728" usemap="#imgmap" />
                <map name="imgmap">
                    <area shape="rect" coords="48,341,294,275" href="http://www.example.com/">
                </map>
                <!--<a href="#" style="display:block; background:#00F; width:356px; height:66px; position:absolute; left:1px; top:-19px; left: 162px; top: 279px;"></a>-->
            </div>
        </li>
    </ul>
</div>
<!-- <ul class="flex-direction-nav">
    <li><a class="flex-prev" href="#"><i class="icon-angle-left"></i></a></li>
    <li><a class="flex-next" href="#"><i class="icon-angle-right"></i></a></li>
</ul> -->

Notes:

1. The coord="48,341,294,275" is in reference to your screenshot you posted.
2. The src="./test.png" is the location and name of the screenshot you posted on my computer.
3. The href="http://www.example.com/" is an example link.

You can auto generate Image map from this website for selected area of image. https://www.image-map.net/

Easiest way to execute!

by creating an absolute-positioned link inside relative-positioned div.. You need set the link width & height as button dimensions, and left&top coordinates for the left-top corner of button within the wrapping div.

<div style="position:relative">
 <img src="" width="??" height="??" />
 <a href="#" style="display:block; width:247px; height:66px; position:absolute; left: 48px; top: 275px;"></a>
</div>