Omphaloskepsis is the practice of figuratively contemplating one’s navel as an aid to meditation or introspection, hence introspection itself.
It’s relatively new, having been invented by Aldous Huxley in his book Those Barren Leaves in 1925. It turns up in only a few dictionaries and seems to be a word that survives more for the opportunity to show off one’s erudition than as a real aid to communication. If so, this article is a further perpetuation of its unreal status. It has been mocked:
A word meaning contemplation while gazing at the navel, omphaloskepsis would be of use only to a deipnosophist. And it has no more business appearing here than has deipnosophist.
The Careful Writer: a Modern Guide to English Usage, by Theodore Bernstein, 1965. I’ve written separately about deipnosophist.
It’s formed from two Greek words, omphalos, “navel, boss, hub”, and skepsis, “the act of looking; enquiry”. The former turns up in words such as omphalotomy for the cutting of the umbilical cord, in the related omphalopsychic for one of a group of mystics who practised gazing at the navel as a means of inducing hypnotic reverie, and omphalomancy, an ancient form of divination in which the number of children a woman would bear was determined from counting the knots in her umbilical cord at birth.