Make use of some of the resources on the tabs below (carols, reflections, scripture, literature, etc.)
Buy and set off fireworks to mark the beginning of the New Year.
If you have a pig, walk it on a leash (a tradition from Vienna). If you don't have a pig, don't steal one. Walk the dog instead.
Compile a list of books you would like to read and projects you'd like to do in the next year.
Spend time looking back over a calendar or pictures from 2018 and talking about your favorite moments from each month.
Invite friends or family over and make a black-eyed peas dish called "Hoppin' John's," as is traditional in parts of the American South.
Write a wish for the upcoming year, burn it, then (if you're especially hardcore), drink the ashes in your New Year's beverage of choice, as is traditional in parts of Russia.
Have a discussion: Look up other New Year's Eve/St. Sylvester traditions and have a conversation about the nature of luck. To what extent is the idea of luck compatible with Christianity (or with whatever beliefs, formal or informal, that you hold)? Can you make any sort of a case for it? What is the difference between luck, fortune, and chance?
I like this idea from catholicculture.org: "On the eve of the civil New Year the children may join their parents in a holy hour, in prayer and thanksgiving for the gifts and benefits which God has given them in the past year, and to pray for necessary graces in the forthcoming civil year."
Give one of the gifts you did not give on the first day of Christmas.
Give someone twelve envelopes labeled with the twelve months of the year, featuring a wish, a scripture, a picture, a quote, a poem that they can read the first day of each month this year (obviously, this is also appropriate for New Year's Day).
NO CASH OPTION: Create a song, poem, or picture featuring what you think was the most important memory shared between you and the gift's recipient over the past year.