For a long time I had a huge problem with the way most people say prayers. The use of words like "thou", "thee", "thine", etc., seemed excessively archaic to me, and thus very unfitting for a conversation with your heavenly parent. They reeked of vain repetition, lacking all of the sincerity that I would normally use in prayer. However, I was wrong. It turns out that using such Elizabethan language in your prayers helps it become more effective, and here I will attempt to explain why.
I've had several Sunday School and Seminary teachers tell me that the reason we use such language in prayers is for respect. As a matter of fact, this is true, but not in the way you would expect. For most of these teachers openly had in mind the kind you would have for authority, like using "Mr." or "Mrs." in front of a teacher's name in elementary or secondary school. It turns out that this interpretation is dead wrong. You see, "thou", "thee", etc., are supposed to engender an entirely different kind of respect. But first, let me introduce you to someone:
Martin Buber was a Jewish philosopher who lived from 1878 to 1965. He is arguably most famous for his distinction between "I-It" and "I-Thou" relationships, explained in a book entitled Ich und Du, or I and Thou. The premise of this idea is that an I-It relationship involves someone's encounter with a conceptualization or image of another person, while an "I-Thou" relationship pertains to two beings whose essences meet, so to speak. When I am "I" and you are "Thou", there is no qualification or pretense between us, only two individuals who see each other as they are.
Now, if a respected scholar like Mr. Buber decided to use the word "thou" to describe such an intimate personal relationship, how on earth can we claim that "thou" is a term used to characterize authority? In short, we can't. And it isn't just Martin Buber that uses the word this way. This connotation can be found in word history as well, as demonstrated by this selection from the Online Etymology Dictionary:
"The plural [you] at first was used in addressing superior individuals, later also (to err on the side of propriety) strangers, and ultimately all equals. By c.1450 the use of thou to address inferiors gave it a tinge of insult unless addressed by parents to children, or intimates to one another."
In other words, "thou"'s original meaning is so unlike our normal associations that, except in cases of very intimate relationships, it was actually used to address inferiors. This type of language usage will probably be very familiar for students of French, Italian, Spanish, and other such tongues, as each uses two different types of pronouns to address two different types of people: the formal and informal. They use the formal to address their social betters, people who are older than them, etc., but they use the informal for more relaxed, friendly situations. And, for example's sake, in French, Italian, and Spanish this informal "you" is "tu". Looks familiar, eh?
All this leads up to a very profound conclusion. If the editors of the King James Bible and Joseph Smith used the word "thou" in human prayers to God, it means that, in their respective works, God is inviting us to treat him as our equal. It does not by any means indicate that he is our equal, but rather that he condescends to our level, asking to be treated as an intimate relation, as a friend. In short, the reason why it is so important for us to use "thou", "thee" and "thy" is because such usage is a deliberate act of connection with God, an act of respect in which you abandon all social barriers between he and the pray-er, where you meet each other authentically and openly.