You're right, it's related to jQuery.
[Not really relevant technical stuff] Basically the problem is that jQuery is included twice. A WordPress installation includes a copy of jQuery and jQuery UI and ImageMapper is using them. Unfortunately Accordion Image Menu isn't using the copy included with WordPress but its own copy of the two libraries, which causes conflict with the jQuery included in WordPress.
It's possible to make an option not to include jQuery in ImageMapper. However with the recent versions of WordPress it's not absolutely necessary, because WordPress is able to check if its own copy of jQuery is already included and if it is WP won't include it twice. Another problem is that the Accordion Image Menu is using an old version of jQuery and customized version of jQuery UI and I can't really import the current version of the dialog widget in it.
However the reason why the menu wasn't working was that ImageMapper crashed if the jQuery UI Dialog wasn't available. The newest version won't crash anymore, but unfortunately without access to jQuery UI Dialog the popup window feature can't be used. Links and tooltips will work fine though.
Thanks for pointing this out anyways.