George's suggestion is a good pure-LabVIEW one, but it could be a lot to bite off for a relatively inexperienced programmer. Many of the steps involved are somewhat complicated: getting your desired image into the picture control in the first place, determining which portion of the image was clicked, etc. Plus, if you want to display test names as the user hovers over various hotspots in the image, then you are going to have to poll your picture control in a loop, which can also be complicated and will slow everything down to boot.
If you're on Windows and are willing to live with a Internet Explorer Web browser embedded in your VI's front panel, then there's an alternate method that involves less programming overhead. You can literally use hyperlinks and an HTML image map, as you alluded to in your question, and have LabVIEW react to various clicks that the user makes in the image map by launching your test VIs. Another advantage is that the IE browser will display "tool tip" text as the user mouses over the different image hotspots without generating any extra overhead in LabVIEW.
I've attached an example (Image Map UUT Tester.vi) that shows how to do this (LabVIEW 7.0). I think it will be fairly easy for you to modify for your own purposes.
Let me know if you have questions,
John