"scrawl aimlessly," 1935, perhaps from dialectal doodle, dudle "fritter away time, trifle," or associated with dawdle (which might be the source of the dialect word). It also was a noun meaning "simple fellow" from 1620s.
LONGFELLOW: That's a name we made up back home for people who make foolish designs on paper when they're thinking. It's called doodling. Almost everybody's a doodler. Did you ever see a scratch pad in a telephone booth? People draw the most idiotic pictures when they're thinking. Dr. Von Holler, here, could probably think up a long name for it, because he doodles all the time. ["Mr. Deeds Goes to Town," screenplay by Robert Riskin, 1936; based on "Opera Hat," serialized in American Magazine beginning May 1935, by Clarence Aldington Kelland]
Related: Doodled; Doodling.
Doodle Sack. A bagpipe. Dutch. — Also the private parts of a woman. ["Classical Dictionary of the Vulgar Tongue," 1796]