Q:

Why is the Lord's Prayer so common as a translation?

I'm an amateur conlanger, but I've seen multiple instances of people using the Lord's Prayer as a test or a way of showing off a translation of their conlang.  I've even done it a couple times myself.  
Why is this such a common practice among conlangers?

A:

It is a long standing tradition to use the Lord's prayer as a sample text for illustrating natural languages, for instance, the mid-16th century Cosmographia by Sebastian Münster contains the Lord's prayer in Finnish as an illustration of the Finnish language.
It has the advantage of being available in many languages, including otherwise ill-documented or dead languages.
Its disadvantages are the religious bias and the fact, that some very marked constructions tend to occur in this text (in German, there is a relative clause in the second person singular ..., der Du bist im Himmel, ... a really rare beast).

A:

Apart from the historical usage that jknappen brings up, the Pater Noster is simply a short, relatively straightforward, universally well known text. Probably the other most common text is the Tower of Babel story. Again, short & straightforward. Perfect texts for the glossopoet whose language is not yet all that robust either lexically or grammatically.  This one also touches kind of near to the heart of all glossopoets.
I don't see any particular religious bias in either text, so that's a ymmv (and anyway, not a good road to go down, please). As for that 2s relative clause in the German translation, of course, the same kind of rarity is in the English text. Good exercise for the nascent invented language! And good exercise for the glossopoet, having to think about and sort out curiosities of grammar right from the start!

A:

Christians are quite firm in their beliefs and as such many pray the Our Father (Pater Noster) with great devotion. I would be very surprised if a Christian liturgical celebration did not recite the Our Father at their Sunday services. I know many who recite this prayer before meals.
Given the popularity of this prayer, I am not surprised that it is so commonly used as a translation tool. Even some private schools will pray the Our Father before class and that in turn will be said in the language of the class being taught: Pater Noster for Latin class and the Notre Pere for class in French.
It would be interesting to see if some Christian high school would recite the Lord's Prayer in Elvish in a Constructed Language course if one exists.

The Quenya translation is by J.R.R. Tolkien (published in Vinyar Tengwar #43).
Átaremma i ëa han ëa,
na aire esselya,
aranielya na tuluva,
na care indómelya
cemende tambe Erumande.
Ámen anta síra ilaurëa massamma,
ar ámen apsene úcaremmar
sív’ emme apsenet tien i úcarer emmen.
Álame tulya úsahtienna
mal áme etelehta ulcullo.
Násie.

As one can see the Our father is short enough to be memorized over a very short period of time and this prayer has immense value for a certain percentage of the population.
With all due respect let us remember that the Lord's Prayer was taught by Jesus Christ himself. For those who believe that is huge.

