Python comes from Monty Python. C++ is the successor of C. C is the letter after B (another programming language). Haskell is named after Haskell Curry. Ada is inspired by Ada Lovelace. Java is coffee. Lisp is a contrived abbreviation of LISt Processor.
All of these facts can be found on the Wikipedia pages for the languages listed above, and more or less all other programming languages in existence have similar information linked to them—even ML, the MetaLanguage.
But not Julia. 
Because there does not seem to be any reason: https://github.com/JuliaLang/julia/issues/5499.
“It just seemed like a pretty name”, referenced in the linked interview, is a reason—and an acceptable answer to me, if it is true.
I always assumed it was a reference to the Julia Set
And while we are discussing trivia, what is the reason for the weird hats at page 6 of https://github.com/stevengj/18S096-iap17/blob/master/lecture1/Course-overview.pdf?
[EDIT: fancy, not weird is the term I should have used. Thank you @Evizero.]
fancy hats need no context
Here’s a random thought, maybe only remotely related to this. Since the file extension for Julia source code is .jl, why not reinterpret those two letters to signify one of the defining features of Julia — JIT compiler on top of LLVM? 
Which refers to a video at https://channel9.msdn.com/Blogs/Charles/Stefan-Karpinski-and-Jeff-Bezanson-Julia-Programming-Language. Stefan and Jeff answer the question at time stamp 18:54.
Really guys, you would make everyone’s lives so much easier if you just started telling people it was named after the Julia set 
I was really hoping it was named after “Julia” from 1984, but random name is still a good naming convention.