An easier way than explained below is to use a Google Drawing in Google Drive.
Insert>Drive>Drawing
  


This is an example of an image map. Click on the words to go to the appropriate site.

To make links open in a new window you use the <area target="_blank"... attribute. Google sites however strips that tag. The only way I have found to make it work is to create an html page, store it in a public Google Drive folder and embed it into your page using an iframe.



Below example, the code has been converted to a gadget.



Shiawassee Arts Center Shiawassee Regional Chamber
To make an image map work you have to edit the HTML. I used an Image Map program to create the code. 
 
The program I used is MapEdit
 
Here is the HTML of the result:
Note: Hosting these files in Google Drive is being discontinued so I recommend using DropBox instead. You can also host your XML files in a file cabinet page in a Google site.  HTML files will have to be converted to gadgets.  See THIS SITE
 
<DIV style="DISPLAY: block; TEXT-ALIGN: left"><IMG src="http://sites.google.com/site/k8omsbob/imagemap/imagemap.jpg" useMap=#imagemap border=1></DIV>
<DIV><MAP name=imagemap>
<AREA title="Shiawassee Arts Center" shape=RECT alt="Shiawassee Arts Center" coords=24,5,195,39 href="http://www.shiawasseearts.org/sac.htm">
<AREA title="Shiawassee Regional Chamber" shape=RECT alt="Shiawassee Regional Chamber" coords=76,49,392,87 href="http://www.shiawasseechamber.org/">
<AREA shape=RECT alt="" coords=0,0,0,0></MAP></DIV>


Creating an image-map (from "prac")

More info  


HTML page uploaded to Drive and inserted using iframe:

<html>
<head>
  <title></title>
</head>
<body>
<div style="display:block;text-align:left"><img border="1" src="https://sites.google.com/site/k8omsbob/imagemap/imagemap.jpg" usemap="#imagemap"></div>
<div><map name="imagemap">
<area target="_blank" alt="Shiawassee Arts Center" coords="24,5,195,39" href="http://www.shiawasseearts.org/" shape="RECT" title="Shiawassee Arts Center">
<area target="_blank" alt="Shiawassee Regional Chamber" coords="76,49,392,87" href="http://www.shiawasseechamber.org/" shape="RECT" title="Shiawassee Regional Chamber">
<area target="_blank" alt="" coords="0,0,0,0" href="http://www.k8oms.net" shape="RECT"></map></div>
</body>
</html>

Gadget code:

<?xml version="1.0" encoding="UTF-8" ?>  
<Module> 
  <ModulePrefs title="imagemap" />  
  <Content type="html"> 
     <![CDATA[  
<div style="display:block;text-align:left"><img border="1" src="https://sites.google.com/site/k8omsbob/imagemap/imagemap.jpg" usemap="#imagemap"></div>
<div><map name="imagemap">
<area target="_blank" alt="Shiawassee Arts Center" coords="24,5,195,39" href="http://www.shiawasseearts.org/" shape="RECT" title="Shiawassee Arts Center">
<area target="_blank" alt="Shiawassee Regional Chamber" coords="76,49,392,87" href="http://www.shiawasseechamber.org/" shape="RECT" title="Shiawassee Regional Chamber">
<area target="_blank" alt="" coords="0,0,0,0" href="http://www.k8oms.net" shape="RECT"></map></div>     ]]>
  </Content>  
</Module>