In Japan, the first dream of the new year is believed to set the tone for the kind of year it'll turn out to be. Since New Year's Day is a day for quiet celebration, and people don't start returning to their daily routines until the second day, the hatsu-yume (first dream of a new year) is usually the dream you have on the night of January 2.
The importance people attach to hatsu-yume seems to go back to ancient history. For instance, there's a historical document that refers to a hatsu-yume dreamt by Emperor Suinin, who is said to have reigned around the fourth century.
Legend has it that the three "best" dreams you can have are about Mount Fuji, hawks, and eggplants - in that order. There're a lot of theories explaining why these three should be considered so auspicious, one of them being that it's about relative height. The tallest mountain in Japan is Fuji, near which is another mountain called Ashitaka (-taka means hawk) that's about half as tall as Fuji. Eggplants were added, people think, to poke fun at their high prices in ancient Japan.
The "first dream" was no laughing matter for people in the feudal period, though. They went to great lengths to make sure they had one of the good dreams - one way being to put under their pillows a drawing of a ship of treasures with the kanji (Sino-Japanese character) for treasure written on its sail. This became a common practice around Muromachi period (the fourteenth to sixteenth centuries), with people from all walks of life - from the most powerful military rulers to the common townspeople - sliding a drawing of a treasure ship under their pillows in the expectation that the year to come would bring them greater joy and prosperity.