NY source income is taxable to NY state. NY taxes telecommuters on NY income so in or out of the state, you have NY income and pay NY tax.
NJ drivers license, MN mail, it sounds like you were passing through NYC. You were in NY for less than 184 days so you are not a resident of NY. You can file a NY part year return and NYC would be for the time actually living in NYC.
Home is where the heart is, where you intend to stay. Is NJ home? If yes, then file a NJ return. If no, don't file a NJ return. NJ may want you to show that you moved on since you have a NJ license. NJ may feel that you are a NJ resident. You can show them your NY part year return to prove you left NJ - but was it a temporary absence or moving?
MN- you were present in MN, was it passing through just visiting family or is that home? Visiting a state does not require a return. No license, no voting, MN would not know you were there.
Back to you, where is home - you kept your NJ license, makes me think.... NJ
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