The Talmud and Midrash state: "Fire is 1/60 of hell, honey is 1/60 of the manna, Shabbat is 1/60 of the World to Come, sleep is 1/60 of death, and dreams are 1/60 of prophecy. Dreams are the buds of prophecy."
Some commentaries say the above are all based on the rule that non-kosher food which gets mixed into a kosher food is annulled in a ration of 1/60, assuming that it is not sour, salty, bitter or spicy. This is related to the taste threshold of the average human (see Pfieffer's Handbook of Physiology). Accordingly, something which is on the threshold of existence but not quite "there" is called "one sixtieth."
Maimonides states: "As you are aware, our Rabbis state that a dream is one sixtieth of prophecy; and you know, that it is inappropriate to make comparisons between two unrelated concepts or things...and they repeated this idea in Midrash Bereshet Rabbah and said, 'the buds of prophecy are dreams.' This is indeed a wonderful metaphor, for just as a bud is the actual fruit itself that has not yet developed fully, similarly, the power of the imagination at the time of sleep is exactly that which operates at the time of prophecy, in an incomplete and unperfected state."
There is a mystical idea behind one sixtieth (at least regarding dreams) which is based on the statement in the Zohar which states "There are six levels [each one encompassing ten sub-levels] between netzach (eternity) and ratzon (will)." Therefore dreams, which have their source in ratzon, because they are of the world of freewill, are one sixtieth of prophecy which is from the world of netzach.
- Talmud Tractate Berachot 57b
- Midrash Rabbah Genesis 17:7
- Maimonides, Guide for the Perplexed 2:36
- Zohar Pekudei 254a