Okay, I tried the latter two methods for turning off hints: setting AFNI_HINTS to NO or KILL, and using Datamode->Misc. Both of these methods succeeded in turning off hints, but the hanging problem did not go away. Instead, it now hangs when there *would* have been a hint if hints were turned on. What this means in practice is that it hangs if you select a popup, don't make a choice, and then release the mouse while it is over the name of the popup. That is fairly easy to live with: you just have to remember to move the mouse away if you change your mind about selecting something from a popup.
It is possible that compiling with -DDONT_USE_HINTS might solve the problem, but I'm not going to try that until I have a independent need to recompile.
However, while trying these things, I did discover something else useful. Bob Cox observes on the cygwin page that "Apparently you can't set a Toggle button on a popup menu in LessTif." This prevented me for a while from using Datamode->Misc to turn off hints. But after playing around for a while, I discovered a way to set toggles in popups. First, click on the menu title. The menu pops up. Then click on the toggle you want to change. Nothing happens (that's the bug). Now drag on the menu title (keep the mouse button held down). The menu pops up. Drag down to the toggle and release the mouse. The toggle state changes. You have to do it twice like this. It's not pretty but it works...
I hope that AFNI keeps stumbling along under cygwin for a couple of years. Next time I buy a laptop it will probably be a Mac, given the success of OS X, but for now some of us need to make do with what we have...
Regards,
Stephen