What will the NYSE spot price for silver be at market close on January 3, 2019?

There is no way to accurately state this as it lies in the future. Thus the best one can do is assess the factors that affect spot prices. As of 28 December market close, the spot price for silver was approximately $15.40. While the 30-day prices have been trending slightly upward, the annual price trend has been somewhat downward.
Historically, silver tends to move with other commodity prices, and thus as commodity prices trend upward in general, one can expect silver to join the movement.
Macroeconomic trends such as inflation also affect silver pricing. In particular, as economies improve and people have more disposable income, silver prices rise. Since silver jewelry is a luxury item, it is sensitive to people's sense of economic well-being. On the other hand, precious metals are sometimes seen as "safe havens" in times of economic uncertainty.
As markets have been extremely volatile recently, in part due to political factors, this is a fraught time for predictions.