It deals the additional damage per attack under the conditions mentioned, not per die though. If you select this ability 5 times as a 5th level rogue your sneak attacks will deal 3d6+5 damage (under those circumstances).
The goblin rogue favored class bonus says : "Add a +1 bonus on the rogue's sneak attack damage rolls during the surprise round or before the target has acted in combat." - How would you interpret this: per sneak attack dice, or on a total entire round action?
Per sneak attack. So, in the surprise round it's only once. In the first round of combat before your enemy goes, it's once per attack so if you're using a bow or something then you can get quite a few of them.
I think others may confuse you with their answer a little, to clarify it is not one per sneak attack die. That would be ridiculous. It would be once per attack that qualifies as a sneak attack so long as it is in the surprise round or before the opponent has acted in combat.
Let's be explicit. Say you're a 8th level goblin rogue with a typically insane sneak (goblin rogues should have sneaks at ridiculous numbers) and you're using rapid shot with a short bow at close range.
Assume you achieve surprise in the surprise round. Your single sneak attack will do 1d4 damage (base for your short bow) and 4d6+8 precision damage from your sneak attack.
In the first round, assume you win initiative. You get three attacks (two for your BAB and a third for your rapid shot). As you're going before your opponent, he's flat footed so you can sneak attack. all three attacks this round do 1d4+4d6+8 damage.
Now, assume that you've moved in close in like the 2nd round and flank your opponent with a partner. In the 3rd round, you'll get to sneak attack again with your melee but this time you only get your normal sneak attack damage of 1d4+4d6 for each of your attacks.