One dish commonly eaten on the first day of Chinese New Year celebrations is 羅漢齋 (luó hàn zhāi). This is a vegetarian dish, often translated into English as "Buddha's delight" or "monk's vegetables". Many people prefer to stick to vegetarian food on this day, and 羅漢齋 is a delicious way to do this. It's a savoury stew of fresh and dried ingredients, flavoured with red fermented beancurd.
Finding a good version of 羅漢齋 on a restaurant menu can be a little tricky. I have eaten many, many fairly pedestrian dishes listed as "Buddha's delight" or "monk's vegetables", ordered from restaurants and takeaways that serve Westernised Chinese food rather than the real thing — often a sad selection of tinned vegetables (water chestnuts, bamboo shoots, baby sweetcorn, mushrooms) in a gloopy brown sauce, maybe with some fresh carrots and mangetout if you're lucky.
Ordering from a Chinese-only menu is probably safer, particularly if the name mentions specific ingredients that are usually omitted from the Westernised version, such as 粉絲 (fěn sī/glass noodles) and 南乳 (nán rǔ/red fermented beancurd). I had a rather nice version the other week from Joy King Lau in London, which was listed as 粉絲南乳羅漢齋煲 (fěn sī nán rǔ luó hàn zhāi bào) — the 煲 (bào) here refers to its being served in a claypot.
To make luó hàn zhāi at home, check out Sunflower's 羅漢齋 recipe. Note that the ingredients for the dish may vary between chefs and between families; see the Wikipedia article on Buddha's delight for a listing of other ingredients commonly and less commonly used.
As Sunflower mentions, an essential ingredient when serving 羅漢齋 as a New Year dish is 髮菜 (fà cài), known in English as "black moss", "hair moss", "hair weed", and variations thereon. According to Wikipedia, it's actually a type of bacterium. It's sold dried, in which form it resembles long, fine hair, hence the name. This ingredient is prominently visible in the photograph above.
The reason for 髮菜's importance here is that its name sounds similar to the phrase 發財 (fā cái), which means "becoming rich" — note that the tones are the only difference in pronunciation (in Mandarin, both tones differ, while in Cantonese, 髮 has the same tone as 發). 發財 forms part of the traditional New Year greeting 恭喜發財 (gōng xǐ fā cái), which translates loosely as "wishing you prosperity".
Do note, however, that according to Professor Wayne Armstrong of Palomar College, harvesting of 髮菜 contributes to desertification, and is now restricted in China. Relatedly, an article in the Hong Kong Standard notes that there is counterfeit 髮菜 on the market, at least in some countries, and it can be hard to tell apart from the real thing.
Other common 羅漢齋 ingredients are also considered by some people to have auspicious associations; see this article from a Hawaiian newspaper, which lists a number of such associations.