My tree has a branch that hangs over my neighbor's house. My neighbor has made the statement that she will not pay to cut the branch, she will just wait until it falls and then sue my homeowners insurance. Is cutting the branch that hangs onto their property and over their house my responsibility or theirs?

Is the branch alive and healthy or dead?

A branch hanging over the property line is hers to do with what she wants with, but she can't charge you for the cost of cutting it.
(And she wouldn't sue your insurance company; she would sue you. She would lose - unless there was clear evidence that the tree is dead and clearly a danger.)

However, neighbors have to deal with trees on borders all the time, and good neighbors work out a compromise.
Otherwise you could find your tree cut right down the middle, or you could have countless other problems over the years.
If it's your tree and it's endangering her roof, don't you feel any sense of obligation at all? I would hope that ethics would play a part here.

The tree is your and it is your responsibility for maintaining it. If it can be proven that the tree was branch was neglected and you were negligent in trimming it you could be successfully sued.

While she has the right to trim or even cut off the part that overhangs she doesn't have the responsibility to do so.

So if you feel that branch is a danger, then you should trim it before it causes damage you will be responsible for.

The tree has a few dead branches, but the "arm" as the tree cutters call it, is not dead or diseased. I do want to get it cut, because I don't want to have a bad relationship with my neighbors (I really don't want to have any relationship with these). I was just wondering if it was my responsibility. Her husband had the tree cutters look at the branch and he was going to pay to cut the arm, but she refused.

As you can see from the responses here, this is an area of law that goes either way in courts all across the US. Judges get sick of it and tend to make snap decisions. Insurers deny,deny, deny based on anything they can think of.
That's why it's good to offer to pay half, I would say. And I would get the actually dead ones removed while they are cutting that one branch, to save money. Ask for a separate quote for the one branch vs all the others, show it to the neighbor, and offer to pay half - based on the fact that it isn't dead.
Take LOTS of pictures, now and when cut.

Offer to pay half, but if they don't go along, it IS your responsibility to have it trimmed.

A lot more depends on whether the branch is giving flower & fruit or is just a dead branch?

OP has stated that the branch is not dead. Flowers and fruit are not relevant.

She can cut the limb without asking you, anytime they wish, since it is hanging over her property. But the tree is yours and you are liable for damages the tree can do.

Esp now that they have expressed concern over it.

Spoke to the arborist who trimmed my other tree. He checked out the tree in question and he told me that my tree is very healthy. The only dead branch on it is hanging over my yard. He said he's an arborist with 20 yrs experience and he said if the tree arm falls off, he'll come to court and testify that the tree is healthy and did not need to be cut. That's all I needed to know!