I'd bet that the motor runs on 12v.

I'd try a dry-cell battery like they use in industrial battery backup devices.

Good source for all sorts of sizes is batteriesasap.com.

The larger the amphour rating the longer the battery will provide juice to the car. However, some of these type of batteries don't respond well to repeated deep discharging, so you probably need to find something labeled "deep cycle".

You'll need something to charge it properly - the battery backup devices take care of this automatically, but basically all you have to do is put more voltage into the battery than it is rated at. The trick is the right amount amperage to prevent overcharging (and possible bad things like melting, burning, etc.)

Most of those battery powered cars were charged using a wall-wart type charger. So long as the voltage is higher than the rating of the battery (by a small amount - say 2 volts for a 12v battery), it will eventually charge the battery. Slow is probably better than fast - keeps the heat down in the battery.