Ever read the story Stone Soup?
The rough idea is if you don't have a solid idea on how to handle something you pull (trick) others into helping you and eventually the mutual effort of everyone improves the final product. Either you get a fine and delicious soup, or a fun and fantastic roleplaying experience.
In RPG terms I call this style of GMing, the Gumbo Method. Gumbo as a dish is make up of stock, shellfish, thickener, and various spices, meats, and other vegetables to make a spicy dish that is unique to who made it and when it was made. No two pots of Gumbo taste the same, but they have a combined flavor that anyone who knows good Gumbo can identify. As a descendant of Cincinnatian parents I have a similar concept with a regional dish called Cincinnati Chili, a meat stew that has Greek origins. (But I digress.)
I call it the Gumbo Method because unlike Stone Soup, the players will contribute, but there actually is a recipe to this that have ingredients that they willingly or unwillingly add to the mix. The Stone Soup analogy is what I started the ideas as but built on over it through the years. Today, I call the method Gumbo rather than Stone Soup.
The Gumbo Method involves getting your players involved in a game, the beauty of a Gumbo game is you can start the method at any time even during existing games. The first trick is to find the thickener, in this case some RPGs provide you instructions on this. Dungeon World has it during character creation where the PC's establish ties to the other PC's in their character sheet. It's embedded social element of the game, and some players opinions the most important part of character gen.
But what happens if you are already in an existing game of say Pathfinder and you want to 'thicken' up the PC back stories and connections a bit?
Well, one of my favorite methods is what I call a "Sticky NPC". This is an NPC who latches onto the players either via backstory or current events. Sometimes this shows up as a a team pet, a young child or weaker NPC who needs their help. Another is the criminal who knows everyone, including the PCs, or perhaps the lost love who also happens to be family of another PC! All sorts of crazy wacky ideas work, just remember to apply soap opera logic to get them to 'stick'. Yes, you might wind up with the PC's Sister Cousin's Aunt Brother's Childhood Friend Secret Rival, but whatever. If you want to build a Sticky NPC who latches onto everyone in the party, go for it, no mater how crazy it seems. Either the NPC idea works or you pull a surprise twist event that establishes a link between one PC to another.
Another thickener is the common bad guy! Same idea as the Sticky NPC but with negative applications. Sticky NPCs are great, but you can also have common causes, old factions the PC's use to belong to, common home town, whatever works to bond the PC's in casual or bizarre ways. Sometimes weird bonds are the best!
Next you need the meat and stock. The stock is the setting, this is easy. Pick what game world you want to play in and go with that. The Meat is the crisis at hand. Prince is kidnapped, Dragon attacking the city, Kobolds causing problems, your runners need some fast nuyen, there is a space hulk that just warped in near a Empire homeworld. Whatever. You can be as generic and trope tastic as you want to be. In fact, the MORE generic and tropey you are with the primary meat of a game the better. Remember Gumbo has a strong and filling base meat, that's just to give the Gumbo heft. Any GURPS GM who knows how to grab 3 random GURPS setting books and make a combined setting for his players knows you can have really weird stocks and still make it work. So don't get caught up WHERE it's set over what's the problem. K.I.S.S. your primary issue, and set it anywhere. Outer space Disco being attacked by kobolds? Got the books to support it? Sure. Do it.
Now comes the tricky part. The second meat!
That's right, you need to add a second crisis or problem. The best choices here are hostile NPCs. A second faction of adventures, a cult trying to turn people into bugs, Darleks secretly experimenting on everyone during the main crisis. This extra meat is critical for good Gumbo, it adds a switch up to the flavor. It's the sausage to the Shellfish of the dish. 2 pieces of advice to the second crisis, make it subtle (so the PC's don't spot it right away) and make it tangential to the main plot. Like in the cult idea, the cult is capitalizing on the Dragon attack. Or the second faction of adventures lead the goblins to town, etc. It might have been the cause of the first problem, but the primary issue needs to take the limelight in the early sessions or hours of a game.
Once you have a thickened, meaty stew it's time for flavor. Let me tell you right here, there is no better spice to a game than paranoia! Seriously, get the players talking about the crisis at hand. Offer up conflicting clues to the primary crisis and the secondary one, and false flags mixed in. Then let them theorycraft in character or not about what's going on. Make around 60% of their theories true, 30% kinda true, and the last 105 false. You can eyeball the %'s through.
Besides the spice of paranoia, you need the hearty vegetable of rewards. Now, these can be money, gear, npc support, mix it up. Just like good Gumbo has onions and bell peppers added in, mix up the rewards the PC's can gain from their actions.
Once you have the ingredients in the pot, start the game, or keep running it if you are doing this to enhance an existing game. Let the paranoid and Player's banter continue, drop the primary meat (crisis) on them, spice with fear and plotting, slowly simmer the bonding agents of NPC connections and back stories so the PC's gel, finally add the second crisis/foe and put the heat on. Playing the game is cooking.
And the flavor, and the fully tummy? That's after you finish the session with happy players. And if you did it right you'll have weeks of leftovers for later session.
Isn't Gumbo Gaming grand?