Bare Minimum Prep: Names, Numbers, Needs
I told myself I wasn't gonna post today. I need to prep my game next Thursday, I've got a town and dungeon that needs to make it to the table, but god fucking damn ktrey's "d100 - Brainfood for Burgeoning Blogs" prompts are fantastic. Here's what I rolled:
What does the "Bare Minimum" for you to run a game look like? Conveniently, I can break this down into 3 alliterative categories. Funny how that works out, isn't it?
Names
I do not like to cheap out on my NPC names. Sure, your old fantasy name generator can work in a pinch, but my approach to names is a bit different. I have strong associations between names and personalities, likely shaped by media and historical figures.
To me, an NPC name acts like a banner signifying the personality I should give them. A Theodore will not act like a Randall, a Susie will not act like a Brunhilde. If you use fantastical nonsense names, pay careful attention to the types of sounds you use; Xylvaris conveys someone/something much different than Grukhh.
When naming locations, feel free to use something obvious and stupid. The forbidden forest of my hexcrawl is called the "Verfallen", a painfully overwrought portmanteau of "Verboten" (German for "forbidden") and "Fallen".1 The real world is full of painfully obvious place-names, and intuitive names increase game-ability. You know you're in for some rough shit when you enter Blighttown2
Numbers
This one might be optional in your more narrative-focused games, but for what I run I always need a nice set of numbers at the ready. If your system has a bestiary, then you're good to go! "Proxy" anything unusual or original to your world by swapping out a few numbers, or just use bears. If you're unlucky enough to be running a system without a bestiary, generate a handful of generic templates; I recommend a generic "mook" that's average in all stats, and a handful of specialized "elites" such as your brutes, casters, and slippery little tricksters. Once you have those frontloaded you should only have to prep stats for your big boss monsters individually.
Needs
You could technically run a functional game with just names and numbers prepped, but your NPC "wants" or "needs" are what will drive your game forward to somewhere interesting. Conflict happens when two or more groups have needs that they do not believe can be achieved simultaneously; give enough NPCs some interesting needs and you will eventually end up discovering a conflict.
In general, I think that the closer a characters needs are to the bottom of the Hierarchy of Needs, the more likely the conflict is to get violent. A goblin stealing food from the town's storages in a dreary winter will probably result in more bloodshed than the baron's wife having an affair. One isn't worse than the other, but if you find a conflict unlikely to fit in the tone of your campaign, try shifting the NPC needs up or down the pyramid.

Whether or you want your NPCs to be a powderkeg of tension waiting for your players' actions to set off a catastrophe, or you want one or more NPCs to make a strong action towards their needs and force the players into reaction is up to you and your group.
Edit: This table from new blogger Meticulous Mezzanine is a fantastic way to get some material needs into your locations. Give it a shot!
tl;dr
There's a lot more I actually like to have on hand to run my games, connective tissue like maps and procedures, more details on how NPCs are likely to behave, but if I were to really boil down my prep to the bare necessities, the core of it really lies in the NPCs. A name to remember who's who and inform my roleplay, numbers for any NPC who will interact with the formal rules of the game (combat, typically), and needs to inform conflict and give some direction to the evening's play, is all I really need to entertain my friends for a few hours. And that's likely all I'll have for next week's game if I keep blogging like this.