In 4th edition dungeons and dragons, when you use the daily power of a Bilethorn Weapon (Adventurer's Vault, p, 64), does it repeat the damage dealt by sources like sneak attack and poison, or just the damage from weapon enhancement and stat bonus?
Yes, Bilethorn Weapon's power copies any extra damage*.
*As long as it's actually extra damage, and not a separate damage roll.
From the Adventurer's Vault, p64:
... Use this power when you hit with a melee basic attack. At the start of your next turn, your target takes the damage again, but all of the damage is poison.
It doesn't say that they take only the weapon & enhancement damage again; they take the full damage dealt by the attack.
From the PHB, p117 (emphasis mine):
Once per round, when you have combat advantage against an enemy... an attack you make against that enemy deals extra damage if the attack hits. ...
Sneak Attack isn't its own separate "packet" of damage; it's part of the damage of the attack that triggered it.
(Note that the Sneak Attack feature of the Thief, from Heroes of the Fallen Lands, is functionally identical in wording, and so the same logic applies.)
From the PHB3, p65 (emphasis mine):
Stone Fist Flurry of Blows
Trigger: You hit with an attack during your turn. ...
Effect: The target takes damage equal to 3 + your Strength modifier. ...
Consider a monk's Flurry of Blows, on the other hand. Where Sneak Attack added extra damage to the attack that triggered it, Flurry of Blows deals its own damage. A Bilethorn weapon would not copy any damage dealt by Flurry of Blows associated with the triggering attack, because it only copies the attack dealt by the triggering MBA, not any other "packets" of damage.
TLDR: Bilethorn weapon copies any damage described as "extra" or "bonus" damage. It does not copy damage from effects that dealt damage on their own, rather than modifying the triggering attack's damage, even if they were triggered by the same attack.