The patch on February 2nd introduced new weekly quests doable in ICC 10 and 25 man raids. The reward for completing the quest is a Sack of Frosty Treasures: A bag with some gold, 5 Frost Emblems, and a chance at epic gems or even BoE gear that drops from trash in ICC.

As you progress through the instance you'll eventually come across one of the NPCs that will trigger the quest you'll get for that lockout period. The one you end up with is random and it can't be changed.

You can do these on either 10man or 25man, I've yet to find confirmation of the ability to do them both within a week. I don't feel like that should be possible but who knows.

So the weekly quests you can end up with are:

Deprogramming - Lady Deathwhisper - Can be attempted again by wiping

Overview: One of the adds in the first wave of Phase 1 will be replaced by a mob named Darnavan. He is an enemy but the goal is to make sure he survives the entire fight against the boss.

Details: Darnavan has no more HP than a standard Deathwhisper add and will go down in seconds if focused. He can be snared but is immune to traditional CC such as traps, polymorph, ect. He has Sunder Armor and Bladestorm and should be tanked away from the raid. Having so little health, the player tanking him needs to be sure he only produces enough threat/damage to keep Darnavan's attention and doesn't risk killing him. If you do accidently kill him you can wipe, abandon the quest, and reaccept it to try again.

Residue Rendevous - Rotface AND Festergut - Cannot be reattempted due to the nature of the quest.

Overview: Players need to gather a debuff from Rotface, and then a seperate debuff from Festergut. And then while having both debuffs active you go and talk to the quest NPC to finish the quest. Death, Divine Shield, and Ice Block will remove the debuffs. If a single player meets the requirement for completion when returning to the NPC the entire raid gets credit.

Details: The debuffs don't actually do anything harmful. You get Rotface's debuff by taking one tic of damage from his Slime Spray. You'll get Festergut's by dealing with spores which is a normal fight mechanic so everyone will get it. Note that killing one boss and then wiping on the second means that you have failed because you can't regain the debuff from the boss you killed. Each debuff lasts for 30 minutes when applied.

Update: The theory of having a player hearth out or being DI'd out of combat to preserve both debuffs has been debunked. You will lose the debuffs if you attempt either of these things.

Blood Quickening - Blood Princes AND Blood Queen - Not repeatable

Overview: This quest is like an old school Baron's run from Strath. It starts a 20 minute timer when you enter the Blood Halls section of ICC. From the time you enter you have that much time to clear the trash, clear the Princes encounter, and pull Blood-Queen before she executes her prisoner.

Details: Not much to say here other than trash must be cleared quickly, the princes have to be one-shot, and you have to reach the Queen ASAP. If the timer expires while you are in combat with the Queen she won't execute the prisoner unless you wipe to her. In which case she will kill him just as soon as combat is over. This is perhaps the least forgiving of all the raid weekly quests. Note that you cannot clear trash and then walk back to accept the quest; as killing anything in the blood halls will despawn the NPC and your weekly is lost.

Respite for a Tormented Soul - Sindragosa - Repeatable if failed

Overview: NPC gives you an item to use on Sindragosa when she is "weakened". I take that to mean when she is at low health which means use it during Phase 3. I could be wrong.

Details: Using the clicky items starts a 30 second channeling process. Considering the amount of movement involved in the fight the idea of standing there channeling this thing during phase 3 (which looks like a huge DPS race) seems completely unreasonable. Its been confirmed thus far that having the beam being channeled on her WHILE she dies doesn't award quest credit. I'm looking for more information on this one still.

Update: Apparently the act of channeling the beam on her causes a stacking debuff to build up. The combined channeling of your raid needs to get 100 applications of this debuff on her before she dies.

Securing the Ramparts - Rotting Frost Giant - Repeatable if failed

Overview: Rotting Frost Giant is a miniboss that shows up on the path to the Gunship fight if you keep running past where the frostwyrm shows up instead of taking the route to the ship. Kill him, hand in quest, get badges.

Details: The giant has his own boss mechanics! Quoting Wowhead comment:

"During the fight, the frost giant will periodically (~20 seconds) cast death plague on one random player from the raid (3 on 25 man?). 15 seconds after the buff is applied, it will cause 8k damage to any nearby players. However, if there are no players within range of the player with the debuff when it expires, that player will instantly die. Any players that are in range of the debuff when it expires will also gain Death Plague, and the player that the buff was transferred from will gain Recently Infected, making them immune to Death Plague for the next 35 seconds."

The same poster suggests grouping up your ranged DPS and healers in pairs so that they will 1) Not die instantly to the plague and 2) Hit the fewest possible people with the resulting blast. Aside from this the boss does a knockback. Beyond that he apparently isn't very difficult.

Edit: 2/4/2010 Updates are posted for Rotface/Festergut and Sindragosa.