BREAKING: I've learned the origin (or at least a plausible one) of "copycat" courtesy of Yahoo! Answers.
According to SomnambulistPhew. This had been vexing me for the last three minutes since I wrote this.
The expression may have originated from observing the habits of kittens that learned by imitating the behaviors of their mother. Copycat has been in recorded use since at least 1896, in Sarah Orne Jewett's "The Country of the Pointed Firs" but the expression could be many decades older.