Arexx Mr Basic TR-3: I picked this 4WD platform as the cheapest and still durable platform for my project. The price is about $30 or so. It survived some moderate hits and crashes during my tests.
Li-Pol battery: I decided to go with it just to make runs longer and charging easier. However, the Arexx and many other cheap platforms come with 4xAA or similar battery holder for more usual battery cells like AA. Having them as rechargeable Ni-MH is highly recommended.
KIW-3312S DC-DC step down power converter: If you don't have anything like this, you can easily go with linear voltage converter without losing too much since all low-voltage components drain low power anyway. Something like LM1117-33 will do.
Motor driver: L298N seems to be a popular choice. I also designed Android sketch to support LMD18200. In case of LMD18200, you'll need two of those to control two motors while L298N has all channels in one chip. I purchased an assembled board with L298N to simplify that part of the build.
Bluetooth HC-05 module: the module is cheap enough (about $5-$10) and versatile enough. It's popular too, so there should not be a problem to find it online.
Arduino Pro Mini: any Arduino-based board will do. I like Pro Mini due to their size.
LEDs for head lights: just see what looks cool on your platform and use them.