Past rivalries, Future arguments 

Their rivalry has gone back to very early America. during the early disputes, it was bigger stick diplomacy letting New York have their say. through disputes over their territory. when England bought the Dutch colonies which is where today's New York and New Jersey. New York said that the Hudson River went in around Staten Island and then out to sea. This meant that New York owned Staten Island, Liberty Island,  a few other islands, the whole river, and all ports on the river. 

The two states/colonies with this for a while, but New Jersey was fed up with the division of land and tried taking New York to the Supreme Court. New York ended up not showing up on the court date saying the Supreme Court Had no matter between the states now. You might be thinking that is crazy, but back then it wasn't figured out yet if the power dynamic between Federal and State government.

New York blowing this off did not help. Trying to not cause a fight the US set New York and New Jersey aside and said to settle a deal so they wouldn't force them into the court's reading. the two settled it out by dividing the river in half. In return New York got some of the islands including but not limited are Satan Island and Liberty Island. To New Jersey, they were just happy to get their ports back.

 Around 1980 New Jersey noticed that one of the islands Ellis Island grew in size. When New Jersey brought New York to the Supreme Court about it though the island was New York, but that side of the river was New Jersey. New York stated that the reason it has grown was that they put the dirt from digging the metro there, but when they searched for the documentation they couldn't find it. the court decided that what was originally there was New York's and what was originally under the sea was now New Jersey's. 

This previouse case didn't change much, but it left something intresting. Going off their previous ruling part of Liberty island is owned by New jersey. What makes this interesting is that the area that New Jersey owns has the gift shop. This could convince New York to try to take New Jersey to court to retry the case for in New York.