There is an idea that the first day of the year represents the year, so if you work hard on the new year's day, you might have to work hard through the year.
We also eat "mochi" in soup, which is called "zouni" or "ozouni". You can tell where people are from in Japan by their zouni soup.
http://en.wikipedia.org/wiki/Osechi
photo from wikipedia
http://en.wikipedia.org/wiki/Ozouni
photo from Japanese cooking site "ouchi-de-gohan"
http://www.e-furikake.com/