First of all, is this officially called a Dop Kit or a Dobb Kit? I see references to both spellings in Google.
Secondly, what is the origin of this term for the men’s bathroom kit thingie? Was it some salesman who was forever lionized by his single product, or just some marketing term, or named after the Earl of Dop?
Appy polly loggies if this has been addressed before - my cursory search on “dop kit” didn’t turn up anything (other than “The Straight Dope Too-Cute Kitten Thread”)
From World Wide Words:
The first time I ever heard “Dopp kit” was from my father-in-law; I figured it for a Navy term. Hee hee, now I’ll have to ask him if he knows why it’s called that!
Thanks aseymayo!! Strange that I can buy it online listed as Dobb or Dop Kits…
I wasn’t terribly sure of the spelling, either - but since you didn’t have any luck searching with those spellings, I tried “dopp” at onelook.com and trombone!.. I mean, voilà! There it was!
I’m glad you asked - I’ve wondered about it myself and never thought to look it up until now.
Probably because “Dopp” is trademarked. 
I’ve been wondering about the Dopp kit too.