Actually, around here we don't have a St Nicholas nor Claus. We call him "Άγιος Βασίλης" (St. Basil). I think this is the link. We depict him the same as the rest of the world by now, thanks to globalization of the western world, but I do know the traditional story surrounding the saint and his relation to Christmas (at least the version I remember).
Christmas eve was an important date to the Christian world. The birth date of the Saviour was supposed to carry the message of love to the people.
He was a great figure of the Christian church at his time, an important man. I'm not sure what was the occasion exactly; one version of the story is that the local Roman prefect asked for all the citizens to pay tribute to the empire. People gave family jewels to St. Basil in order for him to deliver them. However, he managed to talk the prefect out of it. However, he then needed to return the jewels to their owners but he didn't know what belonged to whom.
He decided to bake pies and put one jewel in each one. He then gave one pie to each family publicly. Miraculously, each family received the correct jewel.
This story is commemorated and celebrated in the dinner table of the 31st of December, where we bake a pie (more like a cake/brioche) and put a coin or other small trinket beforehand. The cake is cut on the table and the person who gets the coin in his piece is considered to have good luck for the rest of the year.
Greek Orthodoxy celebrates St. Basil in the first of January and this is when St. Basil gives gifts to children, not the 25th of December.
