The state you file taxes for will be dependent on the state of your domicile and where the income was earned.
Your state of residence is where your true, fixed, and permanent home is and where you intend to return. If you are staying with family in New Hampshire temporarily and intend to return to New York, then your residence will be considered New York. If you are with family because you now work in New Hampshire and intend to move out of New York, then New Hampshire would be your state of residence.
Where your income was earned is also important to determine. If you are working remotely in New Hampshire for your employer in New York and your remote employment in New Hampshire is not mandated by your New York employer, a "convenience of the employer rule" will consider your income taxable in New York. If any income was from a New Hampshire employer while you were living in New Hampshire and you intend to move to New Hampshire, you would be a part year resident of both states and only your New York source income will be taxable to New York.
New Hampshire does not have a state income tax, so if all your income is from New Hampshire and you determine you are resident of New Hampshire and none of your income is New York sourced, you won't have a state tax return to file.
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