Here comes round 2 of some ideas, based on Howland's request that we keep scipting to a minimum.
More to do outside Sanctuary that doesn't START in Sanctuary, but perhaps has a few hints about it in Sanctuary:
Walking through Lower, you converse with a beggar and learn he saw a Duergar recruiting for a mission at Fort Mur. Traveling to Fort Mur, the party learns that there is a quest to be done (currently the Mur quests are for higher levels only, and one is unfinished!). This would encourage folks to head out that way.
Another random rumor floats the city concerning some troubel with Kuo Toa harrassing travellers around the Dark Lake. Investigating the problem at the Kuo Toa City, there could be someone who asks you to travel to a VERY dry area of the Underdark (and therefore inhospitable to them) to recover a relic of Blibdoolpoolp from the Formorians/Ogres/whoever who took it. Once there, the party could either negotiate for it or try to take it back by force.
You encounter a lone troll flying a flag of truce (fall into a trap maybe, instead?) and are offered a chance to make some money or LIVE by bringing them the head of a famous troll hunter who is visiting the area and camped out in a nearby cave.
An earthquake creates/opens a new passage to an area that is mineral rich (gems can be found there regularly) but that is dangerous because it has also attracted creatures like Xorn, earth elementals, mephits, and others, maybe even mining parties of Duergar less forgiving than the Grunstubbles.
Another possibility would be to create several MORE areas similar to the "attack the gnolls/urdlenites/beetles" quests in the mines. Make them well outside the regular bounds of Sanctuary and occupying one of the many caves/lairs out there. In short, I envision here something like the old Descent into the Depths of the Earth encounter areas where the players really got a sense of the Underdark as a place with a LOT of life. After all, all those lone gnoll hunting parties I keep encountering have to be coming from somewhere! The party could choose or not choose to enter the area and there would not need to be a particular hook for them to be there (like the old Duergar missing his daughter). Maybe move the existing gnoll caves to that new area and close the existing encounter area - they've moved on after having gotten tired of getting their butts handed to them.
I guess in short, I'd like to see some REASON for my characters to continue exploring the Underdark. Right now, having found much already and determined that certain areas are encounter heavy, interesting stuff low, they have no desire to travel there. Moving things around a bit would bring that flavor back. I know it was kind of fun after the lizard man attack to figure out where everyone had gone! I know having to relocate quests would be much the same.
Finally, I like the idea mentioned by several others of "linked" quests that must be completed in an order. Completing one quest provides the party with a token of some sort that can be used to open the next area - perhaps after a rest or regroup. This would permit the party composition to change a bit if people need to log or whatever. The episodes should be short enough that getting started does not create a 5 hour commitment for you. Perhaps the rewards are compounded such that there is encouragement for completing the series (the item would disappear at server reset).
Over time, I think the DM team could create a library of quest areas that could be rotated into the mix every so often. I understand that creating areas is tremendously time consuming, having played with it myself a bit. Perhaps there could be a call for those of us with developed/partially developed areas to submit them for use/finishing and inclusion in the module.