Report the full amount of the bonus to both states. The main key is that the bonus is taxable in NY. The reason why that's important is because when you report the bonus also to NJ, NJ will give you a credit for the amount of tax you pay on this same income to NY. And because the income was earned in NY, you are taxed there.
Since you moved in the year, you are filing part-year returns to both NY and NJ. However, all of your income is taxable in NY, the first part of the year when you lived in NY as a resident, and the second part of the year when you lived in NJ as a nonresident. You will be asked in the NY return how much of the income you earned in NY was nonresident income. This will be the same amount as the NJ income.
Also, in the NJ income, you will be given a screen that asks About this income. The reason is that program requirements force both NY and NJ income lines to be added together. On this screen, put a check in the NY income box to remove that amount from your NJ return, and then everything should calculate correctly. The screenshot below will assist you with this:
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