What value PTC Thermistor should I choose for over-current protection for a child's electronics kit?
Rather than use a PTC "thermistor" you should use a PTC Resettable fuse. Perhaps that is what you intended? Similar to a glass fuse, a resettable fuse will trip at a specified current and go high resistance to limit the current. It will remain high resistance until the current is removed. Then, after a short time to cool down, it will self-reset and can be used again.

Have a look at these: Bourns

If the highest current from your battery pack in normal operation is 300mA then you need to use a MF-R030. This has a Ihold current of 300mA, so it is specified not to trip with 300mA going through it at an ambient temperature of 23 degree C. It's Itrip value is 600mA, so it will definitely trip with 600mA going through it, although it will take a nearly 2 minutes to do so. With a current of 1.5A, such as a short circuit, it should trip in 3 seconds.