I'm only on my third sandbox so I'm still experimenting with different methodologies.
The first sandbox was very ad hoc because I wasn't really ready to run when my players asked me to start the campaign. (The campaign the other DM was running ended abruptly and I was forced to improvise.) I sketched out the starting area, came up with ideas for the area around it, and came up with a basic premise to get them out the door. Most of my energy was spent on coming up with what was unusual about the world, which was that it was filled with a strange mist that transformed most people into monsters. However, the PCs were part of an exclusive elite group known as Mistwalkers, who for unknown reasons were actually empowered by the mists. So when the town's water source began to dry up, they were asked to find the cause, and forced to venture farther than anyone had ever gone since the arrival of the mists. I perpetually felt like I was behind the 8 ball in that campaign, so I focused on coming up with interesting ideas and seeded them into the world, not really worrying about how they'd be pertinent to adventure per se. It actually worked out much better than I would have ever expected.
The second campaign started with a map. It was a group of islands that resembled the ying yang. I then started erasing and roughing up the land masses until I had a group of islands I was happy with. The history and background grew out of that. Each island was given a rough level range as well as a theme, which then helped me define who and what was on the island. The factions were much less active in this campaign, with most inwardly focused on their own islands. The PCs, having been recently exiled from their homeland alongside other outcasts, were tasked as road wardens due to the acclaim each had achieved in the war that had ended with their banishment. I started giving players a handout each session with jobs that they had heard about as well as rumors. They were free to ignore both, and sometimes they did, but the jobs and rumors were designed to point them in directions from where they could continue discovering things on their own.
With my most recent campaign I did far more prep than I ever had before, and explored a lot of new tools and methods that could help me. I used hexographer to generate a map (basically, I laid out my hexes in continent view and then had it auto generate the kingdom view from that at 6 kingdom hexes for each continent hex). I then used the random tables from the Oldskull Adventure Generator to give every few hexes a theme. Something like Ashen Strath, Howling Orchard, or Chthonic Woodland. These hexes "bled" into the hexes around them to create differentiation. I used Fantasy Calendar to generate a calendar that tracked lunar cycles and weather. And I use the procedures from Into the Unknown to generate gameplay. I've seeded in NPCs and factions that are much more active and will do things on their own, whether the PCs interact with them or not. I've kept doing the handout from the previous campaign, as that was well received.