Modifying AA powered toy engine to be remotely turned on/off
I can't help much on the remote part. You'll most likely want to use radio, but I can't provide anything more specific than that.

I also can't give good advice right now, but that's because of the low amount of information in the question.

So first step would be to carefully disassemble the engine and post some photos of the circuits. 

You'll most likely find one or even two motors inside (they do mention 4x4 drive) and see how they are controlled. Next step would be to decide what you are going to do. The description is a bit vague but I think that they mention multiple buttons on the unit. You'll have to decide if you want to completely switch to remote control, just have remote on-off control and local controls or have full remote and local control.

Basically, if you want remote on/off control, you could accomplish that in two ways: place the extra switch at engine's main power connector or place extra switch on engine's motor power connector.  It could be easier to just switch completely to remote control, but again that depends on internal wiring and available volume. I'd have to think a bit for full remote and local control.

As for the switch itself, to me a MOSFET in TO-92 package looks like a good idea. If you can afford it, get one of the more expensive ones. They'll most likely be around 1€, but will have much lower internal resistance than cheaper ones which means that they'll consume less power for themselves. 

Also for the radio power supply could be a problem. Common voltage levels for hobby electronics are 3.3 V and 5 V. On the other hand, AA battery provides between 1.6 V and 0.8 V, so you may have to make or get prefabricated converter. Of course, this will negatively affect power consumption.