|Click here for related articles on Fanlore.|
Jawn refers to John Watson.
The main reason that people spell John “Jawn” is because Geothebio has popularized spelling it that way. In her doodles, Sherlock is actually sort of stupid, so it’s supposed to indicate that he’s pronouncing John’s name in a silly and affectionate way. In the actual show, Benedict Cumberbatch pronounces “John” pretty normally, but when people want it to sound silly or exaggerated on the internet, they spell it “Jawn” because it’s funnier. I’ve seen a similar thing in the Doctor Who fandom, where “Rose” is spelled “Roes” to be funny. 
"Jawn" was used in very early fanworks and has fallen out of favor.
Jawn was also the name used for the Martin Freeman is a Hedgehog Meme.
- Anonymous: why do they call john ,,jawn? 2012-01-31