I just did a car ride with my three bettas. I used a setup similar to what majesticstorm posted -- but I put the cups into a small Igloo cooler, as I felt this would help keep the temperature more constant.
I put the cooler onto the floor in front of the back seat (right behind my seat), as I felt it would provide a more consistent temperature. If you keep the car at a comfortable temp, they should be OK.
Being paranoid, I also drank half a bottle of water, and kept the half-filled bottle in the cup holder. This allowed me to see what was happening with the water in their cups. LOL. It was actually fine. The water would tilt and move, but didn't splash.
Majesticstorm had it much tougher, switching from between trains and buses (and walking). If you're taking your own car, I think it'll be easier, since it's just one load, and then you can control the temp yourself.
And Fabian's idea is excellent. But I didn't have shipping bags, and I was worried about squashing the bags. I felt the cups would provide structure, and as it turned out, the three cups fit well into the small cooler. Since you only have the one betta, you can pad the cooler with a towel, like Majesticstorm did, to keep the cup from sliding around.