Cirith Ungol in Sindarin means 'pass of the spider' or, more literally, 'spider pass.' Now why would Frodo (who knew elvish, at least to some degree--I think it safe to assume that he had heard ungol before) travel into something called Pass of the Spider? Did he know what awaited him?
Frodo probably thought it was oly a name, and bore no real significance. I would probably do the same.