None of my children have asked this question yet, but I can guarantee you, it’s coming. Sometime in the not too distant future.
Actually, it’s a good question! And I really don’t think God minds that we ask it. The Bible has been copied over and over and over again. We all know how mistakes can happen in these situations. How can it possibly be reliable? 
But how to answer it? It would be easy to get too entangled in the details. In this setting, less is more. Some history is good, but before we get into the history of why the Bible is reliable, we need to set the foundation from the Bible that explains its validity. Here are a few verses that God made sure was in His book that gives us an answer:
II Tim. 3:16 (All Scripture is inspired.)
I Cor. 15:6 (Some of the people who wrote the New Testament saw Jesus in the flesh.)
Mark 13:31 (Jesus stating that His words will never pass away.)
These are excellent verses to get children to memorize, and for the most part, it will suffice until they get older and start thinking about it more.
The Bible is a historically accurate book. The events that occurred in it are not made up. For instance, when Jesus rose from the dead, women were the first to hear about it and a woman was actually the first one to speak to him. (Mark 16:1-10; Luke 24:1-10) If the writers of the Gospels wanted to make something up, this would be it. Women had virtually no ranking in society at that time and it would have been more profitable (humanly speaking) to change the story. But they didn’t. They stuck to the facts. 
This is an internal example. But what about the external (meaning evidence outside the Word of God)? Got Questions.org puts it plainly:
“There are also external evidences that indicate the Bible is truly the Word of God. One is the historicity of the Bible. Because the Bible details historical events, its truthfulness and accuracy are subject to verification like any other historical document. Through both archaeological evidences and other writings, the historical accounts of the Bible have been proven time and time again to be accurate and true. In fact, all the archaeological and manuscript evidence supporting the Bible makes it the best-documented book from the ancient world. The fact that the Bible accurately and truthfully records historically verifiable events is a great indication of its truthfulness when dealing with religious subjects and doctrines and helps substantiate its claim to be the very Word of God.” 
This is only the beginning. Much more could be said on this topic; however, for the audience it is intended for (children), this will prove to give them enough to think about and the confidence that God did not only write his wonderful book, but he also preserved it for us!
1. http://www.soulation.org/media/Q-HowDoIKnowTheBibleWasntChanged.html (Excellent source on this question.)