Good points all the way around. However, let's think about what we're trying to model. The original question referred to sneak attack damage. Sneak attack damage (in this case) is going to be caused by the weapon. I would still tend to say that you should roll sneak attack damage and then apply the minimum damage rule.
Two different scenarios:
Let's consider the weakest Rog 1 (STR 3, -4 modifier; roughly as strong as a rat or weasel) with a mundane shortsword. Let's say he jumps out of the shadows, and shoves his shortsword toward your face and hits. Damage would be 1d6 (weapon) + 1d6 (sneak attack, but still caused by the weapon) - 4 (STR), with a minimum of one. At best you'll do 8 hit points, 2 on average, and 1 minimum. This seems fairly realistic, given the concept behind sneak attack damage (striking vital areas). With a Strength of 3, you're not going to hurt anyone very badly regardless of where you hit them.
The weakest Rog 1, with a Flaming Shortsword (+1d6 fire damage). Let's say he leaps from the shadows and shoves the Flaming Shortsword toward your face and hits. Damage would be 1d6(weapon) + 1d6(sneak attack, but still caused by the weapon) + 1d6 (FIRE!) - 4 (STR), with a minimum of one. At best you'll do 12 hit points, 5 on average, and a minimum of 1. However, fire is fire, and it's going to hurt regardless of how hard you're hit with it. So in this case, I would apply the 1d6 fire damage after applying the minimum damage rule
Does this make sense?