According to Mental Floss, the term jaywalker is derived from the early 20th century American slang term “jay”, which meant a foolish or naive individual. They say that “when such a pedestrian decided to ignore traffic signals and street signs, he was referred to as a “jaywalker.””
The Etymology Dictionary corroborates this, noting that word jay (actually a late 19th century term — 1888 specifically) meant “fourth-rate, worthless”. As for the term jaywalker, it suggests that the word, which appeared no later than 1912* and certainly derived from jay, may have originated in Kansas City and may have had an implied air of “boldness or impudence.”
*Merriam-Webster says 1915. They also tell me that Bartok (whose music I find incredibly painful to listen to — like someone running their fingernails down a chalkboard) rhymes with jaywalk, which though true, is really, really random. I’m at a loss for an occasion in which one would ever rhyme those two words.