Mad-Libs time! 
We’ve got one in the US named Dick Trickle.
I don’t.
[QUOTE=
I don’t think there is a noun that coudn’t mean ‘penis’, in the right context.[/QUOTE]
In one of the Myra Breckinridge novels (Myra or Myron?), Gore Vidal, making a statement, used the last names of the SCOTUS justices for all of his “dirty” words, and used “Rhenquist” for “penis” throughout. Vidal managed a fairly pornographic novel, and you always knew what he meant. “He thrust his throbbing Rhenquist…”
The FBI Special Agent In Charge in Atlanta after the Centennial Park bombing was named Woody Johnson. When he introduced himself at a new conference I had to laugh. I kept expecting him to introduce his partner ‘Dick Morningwood’.
A few years ago I was travelling on the Central Line of London’s Underground and unfortunately sat close to three young (early 20s?) American chaps who were engaging a young English lady in conversation so loudly that I could not help overhear. One of them boasted that he lived on a ranch and that he owned a horse called “Johnson”. He asked the young lady whether she had ever engaged in equestrianism herself. When she said “yes”, he invited her to visit his ranch should she ever find herself in the neighbourhood and promised that she could “ride his Johnson all day long”. At this, his companions could not hold back their mirth.
And that is why you lot call their tadgers “johnson”.
But…isn’t Jimmy for condom short for Jimmy cap? The cap you put on your Jimmy? Or so I thought.
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And “Bob” just seems creepy.
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If you’re jogging toward the bedroom it might be appropriate.
Likewise, there’s nothing you can say that doesn’t sound dirty if you say it right.
“She’s…redecorating her den.” knowing look
"He’s adjusting his carborator. If you know what I mean
“He’s calculating the value of Pi” Enough said.
This is going to be a WAG, but in an effort to actually answer the OP (as opposed to everyone looking for the rim-shot), I’m going to say it goes hand-in-hand (ba-dum-bump!) with the fact that women of the night refer to their clients as Johns.
“John’s Son” could be a sly reference to his ding-ding, and then got shortened to “Johnson”.
But then, it cold be they got called “John” because whole deal centered around their JOHNSON…
So… it may be a chicken or the egg syndrome here… But at least I’m trying 
Old joke:
HER: “Dick, Peter, Rod, Johnson, John Thomas, Willy, Percy… why do men always give names to their penises?”
HIM: “Well, I wouldn’t want all my decisions made by a total stranger, would I?”
[Beavis & Butthead]Heh heh… he said “rim”… heh heh…[/B&B]
Again, in an effort to answer the OP, I’ve long been convinced that an explanation like Frump Jones’ is correct.
“John”, being the most common name in English and American society, was used as a synonym for anything used by “the common man”. Thus, a bathroom /toilet became “the john” (in Shakespeare’s time “The Jakes”, a form of “John”), Prostitutes’ customers were “Johns”, the typical pseudonym was “John Smith” and the legal Everyman was “John Doe” (with runner-up “Richard Roe”, who presumably had a “Dick”)
“John Thomas” as a nickname for one’s private parts is easily understandable, and I’ll bet D.H. Lawrence didn’t make that up himself, but simply appropriated it. From “John” to “John’s son = johnson” is perfectly understandable.
Could be worse. I used to work with a guy named, no joke now, Dick Orifice.
The New York Yankees’ starting pitchers last September included Johnson, Wang, … and Small. I hope they were scouting guys named Tallywacker and Trousersnake to join them.
Is there any connection to the mythical car part the “johnson rod”?
It’s really too bad that Wikipedia deleted the article on Body Parts Slang…it was perhaps the funniest thing I’ve read in a long time…must have been 4,000 terms for body parts.
Upon further review, it appears that a few sites have preserved the list:
Enjoy!
No–don’t! You’ll go blind!
The problem with that hypothesis is that “johnson” for “penis” has been in use since the middle of the 19th century, while “john” for prostitutes’ clients isn’t recorded until 1911.
Maybe, but I think it’s much more likely that the name stuck simply because it’s so generic and anonymous. We use “John” this way in plenty of non-sexual situations. Soldiers receive “Dear John” letters. “John Doe” stands in for anyone whose name is unknown or withheld, “John Q. Public” personifies the average American man, and “John Bull” personifies the average English man. Variants of “John” – “Hans” and “Ivan” are typically used in a similar way to refer to Germans and Russians, respectively, and have been for hundreds of years. Heck, even the word “Yankee,” which was originally applied by Dutch colonists in the New World to their English neighbours to the West, is a corruption of “Janke,” or “Little John.”
We could start from plenty of other non-sexual applications of these names and quickly arrive at the “penis” sense, if we don’t look at how the use actually developed.
We call a toilet a “John.” We might say that this is so because we typically get out our “johnson” on approach.
Samuel Johnson gave us the first English dictionary. We might say that “johnson” is sort of a proto-Cockney rhyming slang: “Johnson” → “Samuel Johnson”–> “Dictionary Johnson” → Dick. We might further argue that his biographer recorded that he was fond of his cat, Hodge, and was always running out looking for oysters for it, and never missed an opportunity to give him a good stroke. Nudge nudge.
There’s a long-standing rumour that Pope John XII met his end at the hands of a jealous husband whose wife he’d been shagging. Maybe this is where the association came from…
…but don’t bet on it. 
Did you call him “Big Pussy” like the guy in the first season of the Sopranos?