Is the Lightning connector safe for my children?

It is totally safe. A Lightning Cable is like a powered USB. In the worst case (an iPad), it's 5 V (and 12 W), far from enough for damaging your children. The fact that if the part touching is more sensible it doesn't mean more damage. Still, if the body part that touches the connectors is more conducting, they might feel a tiny "shock", but more like when sometimes you touch a metallic item and you feel a shock, nothing to do with touching an electric plug.

Whilst this is usually totally safe, there's a few caveats. I realise this isn't electrical engineering SE, but you asked for sound answers so I'll attempt to elaborate.

As others have stated the most important thing isn't the cable but what it's plugged in to. Lighting is a USB standard so we have 5V there, usually up to 2A. Under the best case scenario this will come from a battery pack, forming a closed circuit, you might get a very small tingle but nothing too dramatic as the potential isn't enough to go through human tissue and cause and real damage. You can test this with a standard 9V battery on your tounge, it tickles but doesn't hurt, although I wouldn't recommend long term exposure due to DC bias, but that's another story. If however were talking about a mains power supply, then that is the most important part. A good quality power supply will isolate the low voltage side from the mains voltage, as well as actively regulating the current and voltage passed. In this case it is still safe because earth/ground does not form a complete circuit. In the event of a faulty/low quality power supply however there may not be the electrical isolation between the mains and low voltage sides. This means that although the voltage between + and - is only 5V, there could be mains voltage between the low voltage side and earth/ground or any conductive materials connected to it (appliances etc). As there isn't the isolation, this could result in a nasty mains voltage shock.

That's not the end of the story though, whilst you may get a shock from a faulty/low quality power supply, and it will probably hurt, there are still other devices to protect you. I'll have to just speak for the UK here as I don't know the regulations elsewhere well enough, but all modern domestic circuits have RCD protection. These devises measure the current flowing out vs the current flowing back (simplified) and if there is a difference, i.e. the current is flowing through you instead, they turn the power off. These are calibrated so normally they trip quicker and at a lower current than is required to cause the body issues, but only after the initial painful shock (I speak from personal experience).

In summary, whilst lightening is exposed and there are better, less exposed connectors, if your using a good quality power supply from a reputable manufacturer on an RCD protected circuit, I'd say the risks are minimal.

It should be very safe, if not harmless, for your children.

Reason being, the lightning connector puts a charge though at 5V with a negligible amount of amperage (not going to break out the ammeter to find out).

What causes electric shocks is the amperage, the higher the more lethal. As mentioned in the thread, the amount of volts helps the charge travel easier.

Physics.SE and Electrical Engineering.SE probably have people who understand electricity better than I ever will, but I'll try and give an explanation:

Think of a charge as a car, volts as fuel and amps as speed. If this car were to hit someone at high speed (equivalent to high amps) the person could get severely injured (a similar effect is seen in high-amperage electrocution). However, if the car is travelling at low speeds, the person is less likely to get severely injured. In both cases, the fuel (volts) doesn't really play a very big part in the analogy.

(people better at me in electricity please do correct me if there are any mistakes)

In conclusion I'd say lightning connectors perfectly safe and you should have nothing to worry about.

You should be more concerned about the brick (the one that plugs in to the outlet) rather than the lightning cable itself. Check the output amperage or wattage, the higher the value, the more "shock" you'll get. For iOS devices, those values are quite low. You'll get a shock (more like a surprise), but not severe.