There are two ways to answer the question: yes, solar energy can be used at night - when it's dark where you are, it's daytime somewhere else in the world. Or, if you're talking about 'in the same place', the answer is no, unless that energy is stored somehow. That can be storage in the form of heat (the sun heats the ground, or is used to heat water) or it could be stored in a battery using a charger. If you want to show your child that, get a solar PV charger for your phone and charge it up during the day, then show him/her that the battery still has charge at night.

Or you could give the kid an apple to eat: it has energy (sugars) which is solar energy stored in the apple (a form of chemical energy). Or burn a log in your fireplace - also solar energy.

Yes, but you have to change it to not-solar: batteries, hydro-power (solar generated electricity pumps the water up during the day, runs turbines at night) etc.