What is the average cost of auto insurance for a 16 year old in new york?

This depends on where in NY you live, tickets/accidents on the driver's record, type of vehicle, types of coverage on the policy, limits and deductibles, who the insurance company is, whether the child is a primary driver of any car or is an additional driver and whether there are other discounts available (driver's ed, good student, multi-car, homeowner's policy, etc.)

Check with several insurance companies - including online options - to get an average price that relates to your specific information.