There are some shows with characters named Joey in common. All of those characters seem to be immature and not very responsible. (My Two Dads, Blossom, Full House) I haven't seen much of Friends but that Joey might fit as well.

Joey Donovan (coincidentally played by Joey Lawrence) on "Gimme A Break!" seemed fairly mature for his age. (Of course, he was a child on the show- and he had to grow up quick, due to being abandoned as a 7-year-old and having to fend for himself.
But I think the general principle mentioned in the previous post holds for most cases of characters named Joey on shows.
And how about Joe Longo on "Melissa & Joey" (also played by Joey Lawrence)? I'm not sure though if he's actually called Joey on the show or just Joe- but he's certainly a responsible nanny and housekeeper- often more times than Melissa Joan Hart's character.
So sometimes the stereotype can be broken.
(Incidentally, Joey Lawrence also played a Joe Roman on "Brotherly Love" who was fairly responsible.)