My kids are 5 and 7, and would never make it to midnight! But we do order a special dinner, which we enjoy by candlelight in the dining room (as opposed to our usual suppers in the breakfast nook), on the good china.
After dinner, we play boardgames of their choosing (usually it's something they got for Christmas), and have a nice dessert. We listen to music and just spend time enjoying each other's company. We let the kids stay up until about 9-ish.
Nothing wild or raucous, but the kids enjoy it, and it still gets them to bed at a reasonable hour, and leaves some time for DH and me to have some time together as well.
Happy New Year!