Hussar
Legend
Well, let's dig down a bit and think about how this will play out.The one that they just now created or just now decided works there when they asked the gm to use that route? Those odds would be one hundred percent if the gm says OK.
It's a quantum contact that can be cut loose with no risk to the pc and no obligations expected at any point
Version 1: Players win the Background Roulette (5e) - One of the PC's has the Noble background and turns to the DM and tells the DM that he is now going to see the mayor. There is no challenge here. It's a ribbon for the background. Other backgrounds have other ribbons. Player skips over the bits that the player probably didn't want to deal with in the first place (thus a potential reason why that player took that background) and gets to the stuff that he does.
Version 2: Traditional Play. The player has to meet with the maid to try to convince her to help him. Play ensues. Presumably the group succeeds and the maid fades back into the ether never to be seen or thought of again since the only reason for engaging this particular NPC was because the DM felt that the scene needed a complication that needed to be dealt with. Since the complication has been dealt with, this NPC joins the thousands of other nameless, formless NPC's that are immediately forgotten.
Version 3: Limited player Authorship: The player decides on the spot that his sister works for this particular mayor. The DM now has an NPC with a direct tie to the player, presumably the player will now actually respond to this NPC and this fleshes out the character and the world that the character inhabits. We have a potentially recurring NPC that the group has an actual reason to interact with.
Version 4: Limited player Authorship with mechanics. This is exactly the sort of thing that Action Points in 3e were created for. It was actually a really cool idea that you could spend AP's this way and, with the DM's approval of course, have limited authorship abilities built right into the game. A fantastic reward for good play. Player plays really well, gets an Action Point, spends the action point to mold the setting slightly and introduce interesting stuff, which in turns gets him rewarded with Action points. Fantastic way to do it. Frankly, I'd have zero problem in 5e with spending Inspiration this way.
Now, I know which one I favor. Then again, if your players are so completely out to freaking lunch that creating an NPC on the fly will break your game, I'm sorry, but no amount of advice is going to help this table. @tetrasodium, you have repeatedly given examples of a table that is completely dysfunctional. No wonder you are having so many problems. Dude, you need to find a new group. Honest. It's not the game. It's not the age. It's the people you are playing with.