The custom of eating shortbread at New Year has its origins in the ancient pagan Yule Cakes which symbolized the sun. In Scotland it is still traditionally offered to "first footers" at the New Year. There are several traditions and superstitions for the Eve of the Gregorian Calendar, (the official civil calendar for most of the known world), New Year:
* cleaning the house and taking out the ashes from the fire
* clear all your debts (emotional and financial)
* all in all -- clear out the remains of the old year and star the New Year fresh
As for Hogmanay don't forget "First footing" (or the "first foot" in the house after midnight). To ensure good luck for the entire household the first foot should bring various things depending on culture but the consistent element is shortbread.
The origin of shortbread dates back in Scotland to medieval times. Traditionally, shortbread was baked in a large round and served cut from the center into triangles.
1/4 cup sugar
1 stick butter
1 1/2 cups flour -- originally and best, in my opinion fine gluten free oatmeal, but feel free to use rice flour
Cream the butter and sugar together.
Sift the flour into the butter and sugar mixture.
Shape into a round loaf about 1/2" thick in a round cake pan (larger is fine), lined with parchment.
Refrigerate for at least 30 minutes.
Just before baking make almost complete cuts of edges into the round.
(Another traditional method, 'cut' into other shapes using a cookie press.)
Either way, lightly for the tops of each slice or cookie.
Bake for 15 minutes in a pre-heated 375 degree oven.
Remove and cut into wedges.