Yes, New York will tax the income because your son is still a resident.
He will have to pay the penalty for underpayment if it was his only job, since the exceptions for the penalty does require withholding and/or estimated tax.
The exception for penalty applies if he had 100% of his prior year tax penalty withheld or 90% of current year. (Penalty for underpaying your estimated tax), and you are correct that New York is very strict on the guidelines and in general.
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