When I ReMix, there are a number of factors I take into account. Melody, harmony, song structure, instrumentation, rhythm, and tone. I've seen some comments in this thread regarding the "original context" or whatever of a song. That would fall under the category of "tone."
Now, when a remixer chooses a piece of music to arrange, they dig into the mechanics of what they think makes a song great. Maybe it has a strong melody, and they want to explore. Maybe the chord structure is interesting to them. Maybe the tone of the piece is haunting, and they want to recreate that. It could be any combination of what I listed above, and that will vary from person to person.
For example, I love the fzero soundtrack.
My remix of Mute City preserves the tone of the original.
My remix of Big Blue does not. Have I desecrated the original, because I chose to transfer parts of the melody, harmony, and rhythm to a different setting? I would say no, but that's up to the listener to decide. i personally think that by putting the original melody into a different context, it can give a fresh perspective on the composers work, and really reinforce what a great piece of music it is in the first place.
If you want to arbitrarily decide that the "original context" is the most important element of a composition, then go for it. I think that's retarded, personally, and you'll be hard pressed to find an established musician or composer who doesn't.
Just my two cents.
Jam it back in, in the dark.