We bought an older home built in the 60's. The driveway has been completely broken up in one section because of a tree root growing from our neighbors yard. She doesn't want to remove the tree, but I don't want to repave the driveway and have the roots do the same thing. Is there a way to get rid of the roots so they don't grow back?

If you do something that kills the tree, or hobbles it so that it falls over, you can be held financially liable for it, and that can be very expensive. A mature tree can have significant cash value. So you should hire a licensed arborist at your expense and ask them what can safely be done. Your neighbor is right in that they don't have to do anything. Its a healthy tree that is on their property and is their decision to make. Roots you have to live with. The same with leaves that fall off it.

Branches may be cut to the property line. However, as Smoothy noted, cutting roots with subsequent tree damage may be a liability on your part if you damage the tree.