Interestingly, the "Gilligan's Island hat" is a "tembel hat" (literally means "silly hat") and was the hat of choice for early Israeli kibbutzim.
It's kind of a national symbol in Israel (as are sandals -- give us a break, it's hot). She may be trying to make a political statement that Israelis understand. I certainly took it that way.
As a further aside, the tembel-wearing shlemiel is kind of a cartoony way of showing the "redneck/everyman Israeli" in Israel:
and Mr. Scwartz clearly used the cartoon character "Srulik" as his model for Gilligan.