You could achieve this via PHP or possibly via some javascript trickery.
There's not a "one solution for all" to this request I'm afraid. (at least not one I can think of right now)
The basic idea would be:
using PHP: you have all the exhibits links in the query used to build the menu. In the menu generating PHP template (site/plugin/index.php) you could add the "next exhibit" link and get the necessary link data from the query.
using javascript (and jQuery): you could look for the active exhibit and then get the link information for the next item in the menu list and then dynamically insert/create your link in the menu.
Off course this is just a very general outline and this would either require you to learn/grasp some PHP or javascript. If you're not the programmer type, I would suggest going the javascript way and get your hands dirty with jQuery. This will come handy in more than just this case and it is actually quite empowering when you overcome the first hurdle.
Hope that helps.