My old group had the opposite problem sometimes. In one campaign, we ended up with an NPC perma-drunk street urchin who was too obnoxious for his own good, and just kinda tagged along with the actual party because it meant he could get a free meal as long as they tolerated him. He was meant to give the characters some hints for the campaign and then get chased off or locked up or otherwise removed, but the party basically instantly adopted him, tried to teach him some manners, got him out of trouble whenever he did some Really Dumb Shit™ like groping a city guard or telling a priestess to go f- herself or barfing on the druid's familliar. I think our DM was a little frustrated for a while that the one-off character he pulled out of his ass just to deliver a message ended up becoming a full-time job for him to play, despite his efforts to get the character killed or left behind.

Fast forward a couple of (in-game) years and what was once an annoying little shit is now an resposnible adult running his own tavern, in which our party frequently resides. All is well that ends well, I guess ;)

(still, that DM never again brought orphans into the play, just in case.)