So 3-5 PCs walk into a bar...
Roll d8 here..
1 Free (person or creature)
2 Defend (person, creature, object, place)
3 Destroy (person, creature, object, place)
4 Investigate (person, creature, object, place)
5 Find (person, creature, object, place)
6 Safely transport (person, creature, object)
7 Acquire (this includes capture/kidnap) (person, creature, object, place)
Then, ok--you got a random NPC table, a random treasure table and a random monster table right?
There's your person, creature or object.
The places on your sandbox map have numbers, right? So you should be able to roll a random location where the PCs have to go for this to go down.
Now you can make a million rumors and want ads for your tavern in a few minutes.
Free Princess Leia from the Death Star. Acquire the Ark of the Covenant in Egypt. Kill Colonel Kurtz in his compound in Eastern Cambodia. Investigate a diamond-smuggler in the Netherlands. Transport this ring to this volcano. Acquire the Eyes of the Overworld in a land full of freaks. Seek out new life and new civilizations in the Alpha Quadrant. Capture a serial killer in the Eastern USA. Find The Holy Grail in the whole world.
This sounds facile, I know, but the idea is--if you already have a map and have already stocked it with locations and random encounters, these three elements are all you need to instantly generate a fairly complex adventure. Everything else that happens along the way in most of the stories sketched above is just obstacles the DM throws up (some of which were presumably already on the map), player tactics, and consequences.
Now there are also what I'd call "second order objectives" which would usually come up either because the PCs thought of them themselves as a means to one of the above ends or because a patron NPC has thought them up as a tactic to pursue one of the above ends, like...
Reconfigure (complex object)
Incriminate (person or monster)
Hide (person, creature, object, place)
Confuse (person or monster)
Seduce (person or monster)
...the list could go on and on forever. Point is that a decent variety of adventures (certainly enough to offer your players some interesting choices) can be gleaned from just the first list, including, eventually, everything on the second list.
(Things like 'seduce' and 'replace' are obviously not necessarily just a means to an end if the PCs themselves think up these, but if your PCs are noticing things they wanna seduce or replace or just go see you don't need a rumor/want-ad table.)
*Props to Midkemia Cities for giving me a good starting point for this list.