New Year's Eve is unfortunately not a woman you meet on December 31st (and on New Year's Day you might wish you hadn't anyway), it is short for "evening" as in "predvečer" (there is Christmas Eve and Midsummer's Eve too, and Hallowe'en is the evening before All Saints). There are no names for the days in British or American calendars, because saints' days are not in the Protestant tradition, so there are no name-day celebrations, and Silvester is not a special day (but a special actor, of course).

Saint Sylvester was originally Pope Sylvester the First, who held office between 314 and 335 AD, at the time of the first Christian Roman Emperor, Constantine, who was possibly baptized by Sylvester as well. He was the first pontiff to call himself "Pope" (in the sense of "father"), and he was responsible for having St. Peter's Basilica built at the Vatican. December 31st was the day of his funeral. This information is quoted from Wikipedia, so it must be true.