A good quote for saying "Happy New Year" is a quote by Alfred Lord Tennyson, who states, "Hope/ Smiles from the threshold of the year to come,/ Whispering 'it will be happier.'" The words come from his play, "The Foresters."
The quote, spoken by the character Robin Hood, actually refers to a person's birthday. He previously had questioned why one should celebrate the passing of a year of life, but he answers himself with the word "hope." This same hope makes the coming of the New Year a thing to celebrate. Mankind always hopes that better, happier times are to come.