ANSWER: If I were able to use only one translation of the Bible for children, it would be the King James Version for two reasons.
For the sake of children, the KJV Bible has a few errors in translation but they are minor and they are well known. Additionally, the beautiful, poetic language of the King James Version has proven easy to memorize for me and even with the strange (to me) use of all the verbs ending in "eth" and "est", I find myself able to remember it better than the more easily understood translations like the New King James Version, or New International Version etc.
I find it remarkable that many of the top scholars of 100 years ago learned to read using the King James Version and hundreds of thousands of "ordinary" people did so following its translation ordered by the Queen of England, Elizabeth I.
The only difficulty is with the use of archaic words and grammar, words which may have changed meaning in the last 200 years.
There is, however, a simple solution available to us today which was not available 100 years ago. This solution is the availability of relatively inexpensive Parallel Bible translations. My particular favorite is the KJV / Amplified Parallel by Zondervan, which puts the King James translation and the Amplified line for line in parallel columns on the same page. This volume allows one to read a passage in the King James Version and then to see how the more easily understood translations have stated it.
With the King James Version as a baseline, one can see the flaws in the new translations yet better understand the King James Version. Zondervan also publishes a New International Version, New Living Translation, King James Version and New American Standard Version Parallel. They also produce a New International Version with a KJV Parallel as well. There are probably many other publishers who have similar offerings and perhaps some that may be even more inexpensive.
For any student of the word of God, I would also recommend the purchase of a Strong's Exhaustive Concordance of the Bible, which has every word in the King James Version numbered and related to the original Hebrew and Greek accompanied by Hebrew and Greek dictionaries giving the meaning and pronunciation of the transliterated words. One can then see how a Greek or Hebrew word is translated in many different contexts.
For computer literate children, there are many Bible study programs available such as Quick Verse which offer many different translations, Strong's and many other reference materials. This has the added benefit of the speed of computer searches and the ability to scan the entirety of Scripture for words or phrases when trying to understand a particular subject or extract the exact meaning of a passage. Easy-to-use and free study programs also exist such as E-Sword.