The nice thing about zombie threads, like all zombies, is they never beg for mercy in the first place.
As for the OP, I'm inclined to answer with a big fat "It depends." Assuming this is not the middle of a battle, the paladin should at least hear the goblin out. If it is feasible to take the goblin prisoner and give it a chance to redeem itself, the paladin should do so. On the other hand, Lawful Good does not mean Lawful Stupid; the goblin does not get a free pass to run away and plot an ambush because it grovels a bit. And if the goblin does not in fact redeem itself while prisoner, it may end up facing execution for its crimes...
...assuming it committed any. Much depends on why the party was fighting the goblins to begin with. If the goblins were raiding and killing innocent people, that's one thing. If the PCs are on a quest and the goblins attack them, well, they may have started the fight, but you're intruding on their territory. You should be looking for an opportunity to make peace.
And if the PCs just saw some goblins minding their own business and carved into them, it doesn't matter what you do with the survivor of the massacre; the paladin fell from grace at the first blow.