My Chat Availability seems to be a persistent setting. I have only tried it in one browser (Firefox 28 for Mac) but it seems to remain set to on or off based on my last setting. Even signed out multiple times just to test.
Have never really noticed it before you mentioning it, so not sure what (normal) behavior is for the setting, but it seems to be persistent on this end.
So not sure if it is a Browser issue or rather a glitch that may require a help ticket to get resolved.