The thermodynamics involved might overwhelm the child, but the question is a good one. It shows the child has taken the information in the book and re-applied it to a new problem (the stated question was about freezing and has nothing to do with steam, contrary to a previous answer). He seemingly already understands that the cold water has cooled and frozen (solidified) the lava. He has moved ahead and designed a new experiment to test the differences between the rates of change of two different materials and how much each one would need to change state to a solid.
This is a golden opportunity to explain "heat", rates of change, and state-change in general terms, rather than needing a hard answer.
Disclaimer: I am an armchair scientist/engineer, but I love this stuff (until the greek symbols show up).
The real consideration is how much heat does it take to change the state of each material and how fast it can move. Well, "it depends on a few things", is the answer. But that is not a bummer if you can follow up with some basic and important concepts, which is the real gist of the question, and this leads to a great discussion: thermal energy transfer, states of matter and state change based on temperature. You may or may not want to omit pressure from that discussion, depending on this child's understanding of science. If omitting, then:
Discuss that water has a 100C range between frozen and gas.
Magma has (I don't know offhand, but it is going to be a LOT more -- how hot does it need to be to vaporize?).
So it depends on where the material started:
Each material will gain or lose heat at a certain rate. the temperature in the freezer can be considered constant (if the kid is a genius, let them know that the magma will heat up the freezer a bit, and the water less so, but it will).
This will open the door to the critical effect here: heat transfer.
From there, you can discuss the idea that if the magma starts at 1000C, but gets to its freezing point faster than water that started at some temperature gets to its own freezing point, then the magma will freeze first. But at a starting point higher than that, it will not freeze first. And so forth.
You can use the analogy of two buckets with different sized holes in them. Even if one bucket is bigger to start out, if you make a hole in the bottom that is the size of the whole bottom, it will clearly empty first. If the holes are about the same size, but there is more water in one bucket than the other, then... you get the idea. So there are two factors: how much water (heat) is left to lose before empty (freezing), and how big is the hole in the bucket. Now, instead of water, the thing that is emptying out of each bucket is heat.
Again, the two takeaways here are:
The concepts of heat transfer, and the rates at which each material will transfer heat. Heat always "flows" from warmer to cooler areas. You might also include that air doesn't accept heat very well, but metal is very good at heat transfer. You could demonstrate this by using an insulator and a conductor, maybe some wood, and some metal. Place your hand on both, see which one heats up faster. The metal has a noticeably better heat transfer rate. Important to pause here to note to your child here that the heat left your hand and entered the material (not exactly, but the child is six and this will suffice). Don't keep your hand on the materials too long or the wood will eventually reach the same temperature as the metal (although that is another good experiment that will show that neither material will ever get HOTTER than your hand).
The concept of state change. Some materials change states in a smaller range than others -- the "bucket" is smaller. Liquid is the only bucket, for now. Materials can be super-heated and super-cooled. The intelligent child can be informed about absolute zero, so technically there are three buckets (states of matter which have boundaries to another state or hard boundaries where temperature cannot be changed further), but again, that's a good follow-up for later. It might very well be the next question they ask... "what happens when you keep heating/cooling?" Once you get this far, explaining what heat is is the logical next topic and from there will make absolute zero very easy to explain, and why there is nothing colder than that. Absolute zero AZ can inaccurately be simply described as the place where atomic vibration (the "definition" of heat) stops. Plasma is probably a bit more difficult for a child to understand. I am taking some liberties here, generalizing and glossing over some important scientific details, but give me a break; the child is six years old.
In my experience, inquiring minds like this one don't crave hard, exact answers nearly as much as the reasoning and concepts behind them. There's time for the math later, but understanding heat transfer and state change is the real opportunity. Remember kiddo, you aren't "getting colder" by going outside in the winter, you are simply losing heat. Since your head loses heat faster than your body, that's why you gotta wear this hat.
PS - one liter of magma or water in a thin sheet will cool off much faster than if it is a perfect spherical volume. Again, "it depends on a lot of things". The bucket analogy there, to explain surface area, is that there are more holes in the bucket, so there is more exposure to the colder air (that air contains less heat) at the same time, so more heat will transfer at once.