The Instance Lock Extension feature was added in Patch 3.2.0. In order to allow for parties and raids to progress through instances at their own pace, players are now able to extend a dungeon or raid ID on an individual basis.
How it works
Existing or recently expired IDs can be extended via the Social tab under Raid by clicking on Raid Info. The ID of any instance to which a player is saved can be extended. Doing so will extend the lockout period by the same amount of time as the original lockout (i.e. extending an Ulduar raid ID will add 7 days, a Heroic: Halls of Lightning dungeon ID will add 24 hours, and a Zul'Gurub raid ID will add 3 days to the lockout time). An ID can be extended more than once. An extension can be reversed on an individual basis provided the player does not do anything in the instance during the extended lockout period that would save that player to the instance.
Q & A
Here are some common questions about this new feature, paraphrased from an official Blizzard post. 
- If the Raid leader locks himself into the Raid for an additional week, does it set the flag for everyone on the team?
- If the raid leader extends his ID, it won't automatically extend the instance ID for everyone who was in the raid. Each player will have to extend his or her own ID.
- However, if the raid leader extends his ID inside an instance, when others subsequently enter the instance to join that raid they will get prompted that they will be saved to the extended ID, which they can accept or decline, exactly as when saving to an instance that has already started.
- Do you have to extend the instance before it expires?
- No. You can extend it before it expires, or you can extend a recently expired ID.
- Can you "undo" an extension?
- Yes and no. If you have not gone into the instance since it would have normally reset (since it was extended), you can still reset it and get a new ID. However, if you have already gotten saved to the extended instance again, you'll have to wait out the rest of the lockout period for a new ID.
- Will you know if you're getting saved to an extended ID?
- Yes. It's just like normally getting saved to an instance. When you zone in, you'll get a message confirming you want to get saved, or to leave the instance if you do not.
- If you extend an ID, but don't manage to kill any bosses, are you still eligible to reset the instance and get a new ID for the week?
- This functionality is being considered, but it is unlikely it will be put in place for Patch 3.2.0. So to answer the question directly, no, you can't do this. Once you enter the instance and click "accept", you're saved for the duration of the lockout period whether you kill any bosses or not.
- As of Patch 3.3.3 the above answer is incomplete. Provided you were previously saved to this Raid ID from the previous week, and don't attempt or complete a Weekly Raid Quest in this instance, you are eligible to reset the instance. If you're new to this Raid ID (ie was not present for the first week, but present for the 2nd week), and receive the "Save" popup, you will be saved, and will not be eligible to reset the instance.
- Patch 3.2.0 (04-Aug-2009): Added.