No special tradition. I honestly am not even sure that my family ever did much for New Year's. I have a vague memory of cheap champagne one year. But I can't honestly remember a special dinner. Although it's possible that nothing sticks to mind after 20-some years. Then, again, it's also very possible that by New Year's Eve, my mother had become budget conscious.
Someone else told me of an interesting tradition she has: steak and crab. What makes it interesting is that it goes back 40 years. She was shopping at a budget grocery store. So budget that they actually had the customers write the prices on the things they were buying! (My parents shopped at a store like that. This is, of course, before scanner systems were so common.) And she found $20 on her way out. No one claimed it, so she decided to treat herself to crab. And thus began a tradition.