Once you've selected the icon you want, get into the configuration editor -- System Tools|Configuration Editor, or aptitude install gconf-editor if necessary. From there -- apps|panel|objects.
You will see a series of folders named object_0, object_1, etc. The trick now is to discover which of those objects is your menu application. Believe me, it could be any of them. Each folder contains a similar collection of information. The solution for me was to click thru all the objects watching the "position" key. Since my menu is at the left side of the panel, it was the one with the lowest number in that key. That turned out to be object_3.
From there on, it's a piece of cake. Click the checkbox for "use_custom_icon", then right-click on "custom_icon", select "Edit key..." and enter the full path to your chosen icon. (For me: /usr/share/pixmaps/menu32.png)