The damage is dealt by the same source as the original source of damage. The doubled damage isn't dealt by Lightning unless it was the original source of damage.

If damage dealt by a source you control is being divided or assigned among multiple permanents and/or players, that damage is divided or assigned before doubling. For example, if you attack with a 5/5 creature with trample and it's blocked by a 2/2 creature, you can assign 2 damage to the blocker and 3 damage to the defending player. Those amounts are then doubled to 4 and 6, respectively.

If another effect (or effects) modifies how much damage a source you control would deal—by preventing some of it, for example—the player being dealt damage or the controller of the permanent being dealt damage chooses the order in which any such effects (including Lightning's) apply. If all of the damage is prevented, Lightning's effect no longer applies.