There are many great Buddha statues strewn about across Japan, each with its own history and appeal. So why would I recommend these three? Frankly, I cannot take the credit--these happen to be the three that the Tokugawa Shogunate formally recognized as Japan's representative three Great Buddhas back in the 1700s. Of course, they had good reason for making their decision, and I hope you'll take the time to go visit each of them to see why.
The Well-to-Do Buddha (Nara, Nara Pref.)
The Iconic Buddha (Kamakura, Kanagawa Pref.)
Built under the rule of the Kamakura Shogunate, this Daibutsu became the icon for Great Buddhas in Japan, probably because of the natural scenery surrounding it. Of course, he was not always exposed to the elements--he used to be in a temple similar to the Nara Daibutsu. However, this Buddha's home was twice destroyed by nature (a typhoon and a tsunami), so the decision was made to leave the Buddha as he was. It was a good decision, because there are few scenes more iconic in Japan than the Kamakura Daibutsu beset by cherry blossoms.
The "Handsome" Buddha (Takaoka, Toyama Pref.)
It beats me why he's considered the "Handsome" Buddha, but a Japanese colleague once noted that the Takaoka Buddha has a motto yasashii ("kinder") face, so is more handsome than the others. In any case, he also happens to be the youngest of the Great Buddhas, but that is not to discount the long heritage of metallurgy that led to his construction. Toyama is renowned for its bronze and copper casting heritage, so it's no doubt that they consider their Great Buddha to be the greatest in Japan. First completed during the Edo Period in 1745, the Takaoka Great Buddha was a symbol of the Hokuriku's emergence as a cultural hub for Japan, but this Buddha fell victim to fire. The version that exists today was recast in 1933, and the neatest element of this Buddha is how the city built up immediately around this important cultural symbol.
The Takaoka Buddha may be the easiest to get to from Tokyo. Simply take the new Hokuriku Shinkansen to Shin-Takaoka Station. Change to a local train line and travel to Takaoka station. From there, it is a short walk.