Ask Mr. Smarty Plants

The Lady Bird Johnson Wildflower Center, home of Mr. Smarty Plants,is dedicated to the growth, propagation and protection of plants native not only to North America but to the area in which those plants grow naturally. According to this University of Nebraska at Lancaster article Growing Mandevilla, the plant is native to Central and South America, specifically, Argentina and requires USDA Hardiness Zones of 10 to 11. Obviously, in San Diego County your climate would meet the growing requirements for this plant, so you can't rely on freezing it out. Although this plant is out of our area of expertise, we do have instructions to help you get rid of it. It will not be easy, and you will probably have to repeat the treatment many times, even if the neighbor removes their mandevillea. 

First of all, you need to find where it is emerging from the ground.  Realize that there may be more than one origin for the vine.   When you find where its growing from the ground, cut or saw it in two.   Immediately, paint the cut end that goes into the ground—the one attached to the root—with an herbicide.  Check with a reputable nursery for an effective herbicide.  Using a small paintbrush or a small sponge with a handle to apply the herbicide.  It is necessary to apply the herbicide immediately after cutting the vine since some plants can quickly repair breaks in cell walls and this would prevent the herbicide from being taken up as readily by the plant.  Be cautious about getting herbicide on any desirable plants nearby.  We advise against spraying the herbicide since it can drift onto desirable plants and perhaps kill them.  Also, read carefully and follow the directions for using the herbicide to avoid any dangers to your health.